The best starting point is perhaps the introductory but
comprehensive survey

- Locally finite graphs with ends: a topological approach IIII (R. Diestel), Discrete Math 311312 (201011); PDF of parts III

- Chapter 8.68 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.

- Tangles and the Stone-Čech compactification of infinite graphs (J. Kurkofka & M. Pitz), preprint 2018; ArXiv.

- Graph-like compacta: characterizations and eulerian loops (B.
Espinoza, P. Gartside & M. Pitz), preprint 2016; ArXiv.

- Ends and tangles (R. Diestel), Abhandlungen Math. Sem. Univ.
Hamburg 2017; PDF

- Brownian motion on graph-like spaces (A. Georgakopoulos & 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 & 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 & 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 & D. Kühn), J. Combin. Theory (Series B) 87 (2003), 197-206; abstract; PDF

- Orthogonality and minimality in the homology of locally finite
graphs (R. Diestel & 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 & P. Sprüssel), Topology and its Applications
(2011), to appear; PDF

- Eulerian edge sets in locally finite graphs (E. Berger &
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 & M. Win Myint), Europ. J. Combinatorics 30
(2009), 356-371; PDF.

- The homology of locally finite graphs with ends (R. Diestel
& P. Sprüssel), Combinatorica 30 (2010), 681-714; abstract;
journal
version; extended
version

- Bases and closed spaces with infinite sums (H. Bruhn & A. Georgakopoulos), Linear Algebra and its Applications (2011), to appear; PDF
- Geodetic topological cycles in locally finite graphs (A. Georgakopoulos & P. Sprüssel), Electronic J. Comb. 16:#R144 (2009); PDF
- Duality in infinite graphs (H. Bruhn & R. Diestel), Comb. Probab. Computing 15 (2006), 75-90; abstract; PDF
- On end degrees and infinite cycles in locally finite graphs (H. Bruhn & M. Stein), Combinatorica 27 (2007), 269-291; PDF
- MacLane's planarity criterion for locally finite graphs (H. Bruhn & 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 & M. Stein), J. Graph Theory 50 (2005), 150-161; abstract; PDF
- Topological paths, cycles and spanning trees in infinite graphs (R. Diestel & 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 & D. Kühn), Combinatorica 24 (2004), 68-89; abstract; PDF
- On infinite cycles II (R. Diestel & D. Kühn), Combinatorica 24 (2004), 91-116; abstract; PDF

- An analogue of Edmonds' Branching Theorem for infinite
digraphs (J.P. Gollin & K. Heuer), preprint 2018; ArXiv

- Hamilton cycles in infinite cubic graphs (M. Pitz), to appear
in
*Electron. J. Comb.*ArXiv

- Hamiltonicity in locally finite graphs: two extensions and a
counterexample (K. Heuer); 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), preprint 2013; 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), to appear in Electronic
J. Combinatorics; PDF

- Extremal infinite graph theory (survey) (M. Stein), to appear
in the Infinite Graph Theory
special volume of Discrete Math (2011); PDF

- Ends and vertices of small degree in infinite minimally
k-(edge-)connected graphs (M. Stein), preprint 2009; 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 & X. Yu), SIAM. J. Discrete Math. 22 (2008), 1381-1392; PDF

- Connectivity
in directed and undirected infinite graphs (K. Heuer,
2018)

- Embedding simply connected 2-complexes in 3-space..., (J. Carmesin, Habilitationsschrift 2017)
- Tree-decompositions in finite and infinite graphs (J. Carmesin, 2015)
- Edge
length induces end topologies (T. Rühmann, 2014)

- The
line graph of every locally finite 6-edge-connected graph with
finitely many ends is hamiltonian, (F. Lehner,
Masterarbeit 2011)

- Infinite
graphs with a tree-like structure, (M. Hamann, 2011)

- Gruppenwertige
Flüsse (T. Rühmann, Bachelorarbeit 2010)

- On the homology of infinite graphs with ends (P. Sprüssel, 2010)
- Extremal questions in graph theory (M. Stein, Habilitationsschrift 2009)
- Graphs and their circuits: from finite to infinite (H. Bruhn, Habilitationsschrift 2009)
- Bicycles and left-right tours in locally finite graphs (M. Win Myint, 2009)
- Topological paths and cycles in infinite graphs (A. Georgakopoulos, 2007)
- Der Zyklenraum nicht lokal-endlicher Graphen (M. Schulz, Diplomarbeit 2005)
- Ends of graphs (M. Stein, 2005)
- Infinite circuits in locally finite graphs (H. Bruhn, 2005)
- Infinite
highly connected planar graphs of large girth
(A.Georgakopoulos, Diplomarbeit 2004)

- Cycles, minors and trees
(D. Kühn, 2001)