Difference between Infringement and transgression

The difference is subtle. Infringing often implies a more minor action or incident (although doesn't preclude something more serious). It also allows for lack of intent or for mere carelessness moreso than "transgression" does.

If you aren't careful, you could start infringing upon his property rights.

A "transgression", on the other hand, implies something more serious, and perhaps more intentional:

Intentionally building his garage over his neighbour's property line was a serious transgression of his neighbour's rights.

In many cases the words could be used interchangeably, but a transgression feels more serious than an infringement, in general.

In general, one infringes upon a right, whereas one transgresses a limit or, by extension, a rule.

This distinction is not very strong, though; not seldom will you see one word used instead of the other, even to the point of interchangeability.

The lack of a strict distinction can perhaps be explained by the origins of both words. To infringe is from Latin infringo, "to break"; to transgress is from Latin transgredior, "to step over". In both cases, it is easy to imagine carrying out the action on a rule and thereby violating it. Perhaps the present meaning of infringe lies in the idea that in- in infringo should represent "in(to)", as in breaking into a house or walled garden, even though I do not believe this sense pertains to the original Latin verb.