I have a question. How heavily is this game inspired by Dungeons and Dragons? Should I play Dungeon and Dragons to help me appreciate (and Understand) this game more? If I don't play DD, is there something else I can do?

It is set in the Forgotten Realms world... the Sword Coast to be precise, and it uses Advanced D&D rules, so it is most certainly inspired by Dungeons & Dragons. Do you need to play D&D before you play this? No, but it couldn't hurt. Everything (well almost everything) you need to know is in the game manual.

The DD game manual or BG's manual?

Familiarity with the rules will help you understand what's going on in fights against Magic-Users and how to deal with them, but otherwise you don't need to know anything about D&D.

The BG manual.

I played Baldur's Gate for the first time without having any experience of D&D or the Forgotten Realms and found that playing the game did not require any in-depth background knowledge, though I did read the manual pretty thoroughly and kept it by my side - something which is not so easy if you don't have a paper version.

Arguably the best thing about these classic games (and this applies to all the Infinity Engine games) is the tactical combat, which I believe is based almost 100% faithfully on D&D rules. The manual contains all the information you need to know in order to learn these rules. It's all very pointy-hat-wearing and beard-stroking stuff (THAC0, dice rolls, modifiers etc.) but the game makes learning this stuff a piece of cake in my view. Just bear in mind that even though you can fight battles in "real time" you are really supposed to use the auto-pause feature, which pauses the game under certain conditions and makes the combat effectively turn-based. But I would definitely recommend that a newcomer read through the manual first : it's a joy to read anyway, being an "old school" manual which attempts to immerse you in the gameworld before you even start the game and explains everything you need to know very clearly.

As for the Forgotten Realms, there are books strewn throughout the gameworld which you can right-click on to read (you can do the same with items, spells etc.) This will give you some idea of the world you are adventuring in, but the Baldur's Gate story itself takes place in just one part of this world at one particular time and it is not necessary to read these books, though it will add to the experience if you do.

As the others have mentioned, you don't need to know the pen-and-paper rules to play and enjoy the game. If you're interested, it's definitely worth checking out, but it's not at all mandatory.

Most of the D&D games sold on GoG use either the AD&D ruleset or the 3rd edition ruleset. As official D&D products go neither ruleset is supported anymore, although 3rd edition remains incredibly popular to this day and there is plenty of third-party support and publications (just don't expect it to have the D&D brand). The Baldur's Gate series, Planescape Torment, and Icewind Dale 1 all use the AD&D ruleset. The Neverwinter Nights series, Temple of Elemental Evil, and Icewind Dale 2 are all based on the 3rd edition ruleset.

Physical copies of either edition are pretty hard to come by these days, and command a pretty hefty premium for second-hand copies since they've been out of print for some time. Fortunately due to the way the 3rd edition ruleset was licensed there are still some very complete online resources available. I'm not aware of any similar resource for AD&D, however, so you may have a harder time getting ahold of those rules if you're interested.

Actually you can now by reprints of 1st edition and 2nd edition Advanced Dungeons & Dragons, along with 3rd edition