Because in France, unlike in English speaking countries, embrasser (i.e. prendre dans les bras, to hug) often go along with one or more kisses, and also as you guessed, because of the verb baiser dual meaning increase, with the vulgar one largely prevailing nowadays.
Rem. 1. Cette étreinte s’accompagne souvent d’un baiser (cf. citation de Verlaine), d’où l’emploi extensif (infra), rendu d’autant plus nécessaire que le verbe baiser évoluait vulgairement.
The explanation that appears obvious in the present context is that, associated with the giving of a kiss, there has to be the coming near one another of the two persons involved and that often those persons will put their hands upon one another so as to steady themselves or so as show more affection, which is the beginning of the act of "embrasser" as the initial meaning of the word implies, that is to say in French,
(TLFi) « Prendre entre ses bras en serrant contre soi ».
However, according to what can be elicited from the actual practice of hugging, since that’s what "embrasser" means primarily, the coming into being of this acceptation of the word would stem from the associating of kisses to hugging and not vice versa.
It’ll probably be difficult to get a better insight on the coming into being of this meaning than that that can be got from the TLFi’s remark ;
Rem. 1. Cette étreinte s’accompagne souvent d’un baiser (cf. citation de Verlaine), d’où l’emploi extensif (infra), rendu d’autant plus nécessaire que le verbe baiser évoluait vulgairement. 2. Il n’est pas toujours aisé, lorsque le cont. n’apporte pas la précision, de discerner si l’étreinte s’accompagne ou non d’un baiser.
P. ext. Donner un ou plusieurs baisers (à quelqu’un) généralement en le prenant et le serrant dans ses bras.
1. Along with this hugging there is often added a kiss, wherefrom the use by extension that can be reckoned with below, a usage made that much more necessary that the verb "baiser" was evolving in the way of becoming vulgar. 2. It is not always easy, when the precision cannot be elicited from the context, to make out clearly whether an added kiss goes along with the hugging.
Here is what says Littré, to show that not everybody in the past approved of this "extravagant usage";
Rem. LITTRÉ, condamnant l’emploi de « embrasser » dans le sens « donner un/des baisers », note, à propos de l’expr. embrasser la main (d’une femme) : On lit parfois dans les auteurs contemporains : « il lui embrasse la main. » C’est mal parler; il faut dire : « il lui baise la main. » Embrasser c’est non appliquer la bouche, mais serrer dans les bras.
The vulgar meaning of "baiser" has no central implication but as can be understood from the above, might have contributed to its being replaced by "embrasser" in its "decent" meaning.
Don’t make mistake with the word “Baiser”, it could be a trap :).
You can say “Un baiser”, for mean “A kiss”.
But “Baiser” alone, in common language, can mean vulgary “Have a sexual relation”.
So, “Puis-je te donner un baiser” mean “Can i kiss you ?”.
But, “Puis-je te baiser” mean, almost litteraly “Can i f*ck you ?”
This is true for common (street) language, not in French literature.