How exactly does mitotic recombination help repair damaged DNA?
Note: the technical term is homologous recombination (as DNA repair is actually its primary function), and this answer focusses on lesions affecting both strands at approximately the same level (as a lesion affecting only one strand at a particular location can easily be repaired by using the clean strand as a template), such as symmetrical double-strand breaks (where both strands of the double helix are severed at the same level; commonly caused by [and responsible for the main lethal effects of] overexposure to ionising radiation; used as the example type of damage for the purposes of this answer).

Short answer

The homologous DNA serves as a template for synthesis of new DNA to repair the damaged area.

Long answer (paraphrased from here)

If both strands of a chromosome are severed at or near the same location, special enzymes unwind the broken ends, and other enzymes then clip off a length of one of the strands from each end (the strand that gets clipped off is always the one with the exposed 5-prime end, in case you were wondering), leaving an exposed single-stranded flap on each side of the break.

At this point, other special proteins grab onto these flaps, and take them looking for an intact homologous piece of DNA (that's why it's called homologous recombination), which can be either a sister chromatid (which is guaranteed to be either identical or very nearly so, and is, thus, the preferred solution when possible) or a homologous chromosome (which is always available, except for haploid cells [such as human gametes], but is virtually guaranteed to be not-quite-identical, and, thus, is a less-optimal solution).

With the help of the aforementioned special proteins, one of the flaps then butts into the homologous double helix, base-pairs with the strand it's complementary to (forming a structure known as a Holliday junction [well, technically, seven-eighths of a Holliday junction, as one of the junction's four arms has only a single strand, rather than two]), and kicks the other strand out of formation.  The displaced strand forms a single-stranded loop hanging off to the side (known, unsurprisingly, as a displacement loop, or D-loop), while a DNA polymerase latches onto the bare 3-prime end of the invading strand and lengthens it, using the homologous DNA as a template.1

At this point, things can go in either one of two ways:


In the simpler pathway (the synthesis-dependent strand annealing, or SDSA, pathway), the polymerase simply continues to lengthen the invading strand until enough DNA has been added to the end to bridge the gap between the two broken ends and base-pair securely to the flap on the other end; at this point, the invading strand backs out, allowing the invaded DNA to reassume its double-helical dignity, and the newly-lengthened flap grabs onto the complementary flap on the other broken end.  More DNA polymerases fill the remaining gaps, and a ligase seals the helix back up.  The SDSA pathway is the major homologous-recombination pathway for cells not planning on undergoing meiosis.
In the more complicated double-strand-break repair (DSBR) pathway, the exposed flap on the other broken end also gets in on the action, base-pairing to the exposed D-loop and likewise being lengthened by a DNA polymerase.  When enough new DNA has been synthesised to completely fill the single-stranded gaps, the exposed ends are joined together by ligases, forming a structure with no exposed ends and two complete Holliday junctions.  To turn this back into a pair of separate bog-standard double helices, two of the four strands in each Holliday junction are cut by endonucleases, causing the two DNAs to separate.  (There are rules as to which two strands are cut; either both of the two strands that cross over in a given Holliday junction, or else both of the two strands that don't cross over in said junction, but never one crossing and one non-crossing strand [as this would merely create a new double-strand break at the location of the junction].)  Depending on how the Holliday junctions are severed, the portions of the two chrom[atid/osome]s distal to the original break may or may not be exchanged between the chrom[atid/osome]s; if both junctions are severed on their crossing strands, or both on their non-crossing strands, this distal DNA will not be exchanged, while, if one junction is cut on its crossing strands and the other on its non-crossing strands, said DNA will be exchanged.  Due to its possibilities for swapping bits of genetic material, this is the preferred pathway for cells that are planning on undergoing meiosis (although it also occurs to some degree even in somatic cells).


Here's an image to illustrate the process:



(Image by Emw at Wikimedia Commons.  Blame them for the typo in "Strand invasion", not me.)



1: This is why the 5-prime strand got pruned back, and not the 3-prime strand; all DNA polymerases known to science lengthen DNA strands in exclusively the 5-prime→3-prime direction, and can only add nucleotides to exposed 3-prime ends (exposed 5-prime ends need not apply).