Learn Romanian: El este așa de neîndemânatic încât era să scape paharele. - He is so clumsy that he almost dropped the glasses.

The structure era să followed by the subjunctive is a fixed phrase in Romanian to express an action that was narrowly avoided, translating to "almost did". Notice that era stays invariable here (it doesn't change to match the subject), while the second verb scape is conjugated to match "he".

Doesn't a scăpa usually mean "to escape"? Why does it mean "dropped" here?

You are right that a scăpa often means "to escape". However, when used with a direct object like paharele (the glasses), it means "to drop" or "to let slip" from your hands.

Why do we use încât for "that" instead of că?

When expressing a result or consequence after phrases like așa de or atât de (meaning "so"), you use încât to mean "that". The word că is used to connect simple statements, like știu că (I know that), but not for this "so... that..." cause-and-effect structure.

Could I say atât de instead of așa de in this sentence?

Yes, absolutely. așa de neîndemânatic and atât de neîndemânatic are completely interchangeable. Both are common ways to say "so clumsy".