Folate, or rather its active form tetrahydrofolate (THF), is a coenzyme that carries a 1-carbon group. The carbon group can occur in various redox states: CH3-THF (methyl-THF), CH2-THF (methylene-THF), CH-THF (methenyl-THF) or CHO-THF (formyl-THF). These can be interconverted by a variety of enzymes, and are used by cells to synthesize a number of products, including nucleotides and methionine.
MTHFR (methylene-THF reductase) converts CH2-THF into CH3-THF. The CH3-THF form is the most abundant folate in blood, and is therefore considered the "transport" form of folate. CH3-THF is also the form used to synthesize methionine, in the reaction
CH3-THF + Homocysteine $\rightarrow$ THF + Methionine
catalyzed by methione synthase (gene symbol: MTR). This enzyme requires cobalamin (B12) as a cofactor. Actually, the reaction occurs in two steps,
- CH3-THF + Cobalamin --> THF + CH3-Cobalamin
- CH3-Cobalamin + Homocysteine --> Cobalamin + Methionine
Loss of MTHFR leads to deficiency of CH3-THF, causing methionine synthase to run out of substrate, and homocysteine to accumulate. Lack of the B12 cofactor also blocks methionine synthase, and it should be clear from the reactions above why supplementing with either CH3-THF or CH3-cobalamin can alleviate the problem (at least in theory). I would guess the phenotypes seen in this condition is mainly due to lack of methionine though, rather than excess homocystine, since methionine is required for a variety of methylation reactions in cells.
This is a simplified view. There are many other interactions between B12 and folate; for example, the methinine synthase reaction is needed for cells to take up and store folates, and therefore B12 deficiency can create a folate deficiency as well. See for example http://www.ncbi.nlm.nih.gov/pubmed/6115113 and http://www.ncbi.nlm.nih.gov/pubmed/10466189 for further reading.