Learn Russian: С точки зрения науки это невозможно. - From the point of view of science this is impossible.

It literally means "from the point of sight". The preposition с (meaning "from" here) requires the genitive case, which gives us точки (from точка, meaning point). Then, зрения is the genitive of зрение (sight or vision), connecting them as "point of sight".

Why is науки also in the genitive case here?

It translates the English "of" in "of science". This creates a chain of three genitive words in a row: точки (required by с), зрения (point of view), and науки (view of science).

Could I use an adverb like по-научному to express this instead?

Not exactly. While по-научному means doing something "in a scientific way", с точки зрения науки is used specifically to attribute a perspective or state an opinion. It tells us whose rules or perspective we are judging the situation by.

Why is the short form невозможно used here instead of a full adjective like невозможный?

Because это (this) is acting as a generic pronoun referring to an abstract concept or situation, rather than a specific masculine or feminine noun. When describing general situations like this, Russian always uses the neuter short form, which looks identical to the adverb form.