Could anybody explain me the expressions "to drop the stern" and "to gain the bow". ?

These terms deal (as you suggested) with sailing -- specifically with the planking of boats.

On lapstrake hulls the planks are joined together either overlapping or rabbeted.

At the bow, when the planks from both sides come together, it is desireable for them to be fitted (rabbeted) into the stem without any lap showing. (To smooth the cutting of the water.)

To do this, the rabbets or overlaps are tapered -- they have a "gain" cut into them, they "gain" depth -- which brings them finally together into the stem with the thickness of a single plank.

At the stern, rather than have many narrow planks meet the transom, some planks are "dropped" -- they will be ended some distance before the sterm and adjacent planks allowed to run wide to accomodate them.

Someone is having you on. Neither of those expressions are recognizable. It's like trying to find 50 yards of shore line or keys to the oar locks. Dropping the stern would only occur on a severe wave front and not desirable. Gain the bow? To gain in nautical terms is to approach or take. That would translate as run to the bow, which doesn't make since because an instruction of that nature would be limited to the purpose. "Grab the Sheet", control line gone adrift, "secure the pole", whisker pole has gone adrift. Adrift means loose out of control.

At a guess I'd think it meant something about weight distribution. EG lower the stern and gain a bit more on the bow. Possibly usefull in harsh weather conditions when you dont really want the front digging into the waves possibly?

Google is a wonderful tool...when there's something to find.

"To drop the stern" refers to dropping a stern anchor.

"To gain the bow" from all that could be seen has nothing to do with boating but instead refers to "Odysseus' attempt to gain the bow by hunting down its rightful owner both reflects his unworthiness to bear it and undercuts the supposedly civilized order constituted by the army at Troy."

Now this is not to say that some clever witted nautically inclined individual didn't twist it to make it sound like he wanted it to.

I saw a travel lift operator "drop the stern" on the hard once. Boy was the owner pissed.

Ummmmm, here's just a guess.

Does it refer to trimming out using your outdrives or trim tabs?

I know on my 30 footer my trim tabs are labeled "bow up" and "bow down." When I raise that bow, the stern certainly comes down.