The best starting point is perhaps the introductory but
comprehensive survey

- Locally finite graphs with ends: a topological approach I–III (R. Diestel), Discrete Math 311–312 (2010–11); PDF of parts I–II

- Chapter 8.6–8 of my book, 5th edition (2016/17).

While the survey is more comprehensive (and includes many pointers to what might be interesting to look at next, including countless open problems), it is also written in a less formal style that makes slightly more demands on the reader. The book chapter may help with precise definitions, should the survey be found to be too informal. It also offers a selection of proofs of basic facts, which are typical for this area and make good introductory reading. There is also an older expository text, mostly written around 2002:

- The cycle space of an infinite graph (R. Diestel), Comb. Probab. Computing 14 (2005), 59-79; PDF

The first few sections of this contain a lot of motivation for the topological concepts used in this field, and still have some entertainment value.

- Ends as tangles (J. Kneip), preprint 2019; PDF
- Tangles and the Stone-Čech compactification of infinite graphs (J. Kurkofka and M. Pitz), preprint 2018; ArXiv
- Ends, tangles and critical vertex sets (J. Kurkofka and M. Pitz), Math. Nachrichten 292(9) (2019), 2072-2091; Journal/ ArXiv
- Ends and tangles (R. Diestel), Abhandlungen Math. Sem. Univ. Hamburg 87 (2017), 223–244; PDF
- Inverse Limits and Topologies of Infinite Graphs (B. Miraftab), preprint 2016; ArXiv
- Brownian motion on graph-like spaces (A. Georgakopoulos and K. Kolesko), preprint 2013; PDF
- On graph-like continua of finite length (A. Georgakopoulos), Topol. Appl. (2014), 188-208; PDF
- Dual trees must share their ends (R. Diestel and J. Pott), J. Combin. Theory (Series B) 123 (2017) 32-53; PDF
- Graph topologies induced by edge lengths (A. Georgakopoulos), Discrete Math. 311 (special issue 2011), 1523-1542; PDF
- The fundamental group of a locally finite graph with ends (R. Diestel and P. Sprüssel), Advances Math. 226 (2011), 2643-2675; abstract/ PDF
- End spaces of graphs are normal (P. Sprüssel), J. Combin. Theory (Series B) 98 (2008), 798-804; PDF
- Duality of ends (H. Bruhn & M. Stein), Combinatorics, Probability and Computing 12 (2009), 47-60; PDF
- End spaces and spanning trees (R. Diestel), J. Combin. Theory (Series B) 96 (2006), 846-854; abstract/ DVI/ PDF
- Connected but not path-connected subspaces of infinite graphs (A. Georgakopoulos), Combinatorica 27 No.6 (2007), 683-698; PDF
- Graph-theoretical versus topological ends of graphs (R. Diestel and D. Kühn), J. Combin. Theory (Series B) 87 (2003), 197-206; abstract/ PDF

- Eulerian spaces (P. Gartside and M. Pitz), preprint 2019; ArXiv
- N-arc and n-circle connected graph-like spaces (P. Gartside and M. Pitz), Topol. Appl. 256 (2019) 7-25; ArXiv
- N-arc connected graphs (P. Gartside and A. Mamatelashvili and M. Pitz), preprint 2018; ArXiv
- Graph-like compacta: characterizations and eulerian loops (B. Espinoza, P. Gartside and M. Pitz), preprint 2016; ArXiv
- Orthogonality and minimality in the homology of locally finite graphs (R. Diestel and J. Pott), Electronic J. Comb. 21 (2014), #P3.36; PDF
- Cycle decompositions: from graphs to continua (A. Georgakopoulos), Advances Math. 229 (2012), 935-967; ArXiv
- On the homology of locally compact spaces with ends (R. Diestel and P. Sprüssel), Topology and its Applications 158 (2011), 1626-1639;s PDF
- Eulerian edge sets in locally finite graphs (E. Berger and H. Bruhn), Combinatorica 31 (2011), 21-38; PDF
- Topological circles and Euler tours in locally finite graphs (A. Georgakopoulos), Electronic J. Comb. 16:#R40 (2009); PDF
- Bicycles and left-right tours in locally finite graphs (H. Bruhn, S. Kosuch and M. Win Myint), Europ. J. Combinatorics 30 (2009), 356-371; PDF
- The homology of locally finite graphs with ends (R. Diestel and P. Sprüssel), Combinatorica 30 (2010), 681-714; abstract/ journal version/ extended version
- Bases and closures under infinite sums (H. Bruhn and A. Georgakopoulos), Linear Algebra and its Applications 435 (2011), 2007-2018; PDF
- Geodetic topological cycles in locally finite graphs (A. Georgakopoulos and P. Sprüssel), Electronic J. Comb. 16:#R144 (2009); PDF
- Duality in infinite graphs (H. Bruhn and R. Diestel), Comb. Probab. Computing 15 (2006), 75-90; abstract/ PDF
- On end degrees and infinite cycles in locally finite graphs (H. Bruhn and M. Stein), Combinatorica 27 (2007), 269-291; PDF
- MacLane's planarity criterion for locally finite graphs (H. Bruhn and M. Stein), J. Combin. Theory (Series B) 96 (2006), 225-239; PDF
- Der Zyklenraum nicht lokal-endlicher Graphen (M. Schulz), Diplomarbeit, Hamburg 2005, PDF
- Cycle-cocycle partitions and faithful cycle covers for locally finite graphs (H. Bruhn, R. Diestel and M. Stein), J. Graph Theory 50 (2005), 150-161; abstract/ PDF
- Topological paths, cycles and spanning trees in infinite graphs (R. Diestel and D. Kühn), Europ. J. Combinatorics 25 (2004), 835-862; abstract/ PDF
- The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits (H. Bruhn), JCTB 92 (2004), 235-256; PDF
- On infinite cycles I (R. Diestel and D. Kühn), Combinatorica 24 (2004), 68-89; abstract/ PDF
- On infinite cycles II (R. Diestel and D. Kühn), Combinatorica 24 (2004), 91-116; abstract/ PDF

- An analogue of Edmonds' Branching Theorem for infinite digraphs (P. Gollin and K. Heuer), preprint 2018; ArXiv
- Hamilton cycles in infinite cubic graphs (M. Pitz), Electron. J. Comb. 25 (2018), #P3.3 ArXiv
- Hamiltonicity in locally finite graphs: two extensions and a counterexample (K. Heuer), Electron. J. Comb. 25 (2018), #P3.13; PDF
- Contractible edges in 2-connected locally finite graphs (Tsz Lung Chan), Electronic J. Comb 22 (2015) #P2.47; PDF
- A sufficient local degree condition for the hamiltonicity of locally finite claw-free graphs (K. Heuer), Europ. J. Comb. 55 (2016), 82-99; PDF
- Extending cycles locally to Hamilton cycles (M. Hamann, F. Lehner, J. Pott), Electron. J. Combin. 23 (2016), #P1.49; PDF
- A sufficient condition for Hamiltonicity in locally finite graphs (K. Heuer), Europ. J. Combinatorics 45 (2015), 97-114; PDF
- Forcing finite minors in sparse infinite graphs by large-degree assumptions (R. Diestel), Electronic J. Combinatorics 22 (2015), #P1.43; PDF
- Extremal infinite graph theory (survey)
(M. Stein),
*Infinite Graph Theory*special volume of Discrete Math. 311 (2011), 1472–1496; PDF - Ends and vertices of small degree in infinite minimally k-(edge-)connected graphs (M. Stein), SIAM J. Discrete Math. 24 (2010), 1584–1596; PDF
- Infinite Hamilton cycles in squares of locally finite graphs (A. Georgakopoulos), Advances Math., 220 (2009), 670-705; PDF
- Forcing highly connected subgraphs in locally finite graphs (M. Stein), J. Graph Theory 54 (2007), 331-349; PDF
- Arboriticity and tree-packing in locally finite graphs (M. Stein), J. Combin. Theory (Series B) 96 (2006), 302-312; PDF.
- Hamilton cycles in planar locally finite graphs (H. Bruhn and X. Yu), SIAM. J. Discrete Math. 22 (2008), 1381-1392; PDF

- Connectivity and tree structure in infinite graphs (J.P. Gollin), PhD dissertation, Hamburg 2019; PDF
- On infinite graphs and infinite groups (B. Miraftab), PhD dissertation, Hamburg 2019; PDF
- Eine topologische Erweiterung des Orientierungs-Theorems von Nash-Williams auf lokal endliche Graphen (L. Jannasch), BSc dissertation, Hamburg 2019; PDF
- Connectivity in directed and undirected infinite graphs (K. Heuer), PhD dissertation, Hamburg 2018; PDF
- Embedding simply connected 2-complexes in 3-space..., (J. Carmesin), Habilitationsschrift, Hamburg 2017; PDF
- On the tangle compactification of infinite graphs (J. Kurkofka), MSc dissertation, Hamburg 2017; PDF, ArXiv
- Abstract tangles as an inverse limit, and a tangle compactification for topological spaces (M. Teegen), MSc dissertation, Hamburg 2017); PDF
- Infinite tree sets and their representations (J. Kneip), MSc dissertation, Hamburg 2016; PDF
- Tree-decompositions in finite and infinite graphs (J. Carmesin), PhD dissertation, Hamburg 2015; PDF
- Edge length induces end topologies (T. Rühmann), MSc dissertation, Hamburg 2014; PDF
- Two sufficient conditions for hamiltonicity in locally finite graphs (K. Heuer), MSc dissertation, Hamburg 2013; PDF
- The line graph of every locally finite 6-edge-connected graph with finitely many ends is hamiltonian, (F. Lehner), MSc dissertation, TU Graz 2011; PDF
- Infinite graphs with a tree-like structure, (M. Hamann) PhD dissertation, Hamburg 2011; PDF
- Gruppenwertige Flüsse (T. Rühmann), BSc dissertation, Hamburg 2010; PDF
- On the homology of infinite graphs with ends (P. Sprüssel) PhD dissertation, Hamburg 2010; PDF
- Extremal questions in graph theory (M. Stein), Habilitationsschrift, Hamburg 2009; PDF
- Graphs and their circuits: from finite to infinite (H. Bruhn), Habilitationsschrift, Hamburg 2009; PDF
- Bicycles and left-right tours in locally finite graphs (M. Win Myint), PhD dissertation, Hamburg 2009; PDF
- Topological paths and cycles in infinite graphs (A. Georgakopoulos), PhD dissertation, Hamburg 2007; PDF
- Der Zyklenraum nicht lokal-endlicher Graphen (M. Schulz), Diplomarbeit, Hamburg 2005; PDF
- Ends of graphs (M. Stein), PhD dissertation, Hamburg 2005; PDF
- Infinite circuits in locally finite graphs (H. Bruhn), PhD dissertation, Hamburg 2005; PDF
- Infinite highly connected planar graphs of large girth (A. Georgakopoulos), Diplomarbeit, Hamburg 2004; PDF
- Cycles, minors and trees (D. Kühn), PhD dissertaion, Hamburg 2001