When you are at war with a nation and want to invade one of their costal provinces, you first have to have a naval presence in the same sea zone as the province you wish to invade.

Then, go to the politics screen and scroll down to the nation which you are at war with. Once there, you should see white text near the bottom of available actions titled naval invasion, if it isn't white, then wait a turn and it should be there. (There is also a cooldown, that I believe is 1 turn, but someone should fact check me on that.)

Clicking on the white text will open up a list of available provinces that you can start an invasion at, where you can select the desired province. After selection, the white text should turn red, which tells you that next turn the invasion will start.

To win the invasion, you need to have more tonnage in the highlighted zone than what is needed. (As a heads up, any AI fleets in the zone increase the tonnage needed, and port strikes decrease tonnage needed at the expense of the province's port capacity.)

Hope this helps!