Translation:That restaurant was lively.
I remember that 忙しい it's used with people, like when you say "I am busy" but when talking about places is にぎやか。
I had to look this up because I didn't recognize the kanji but thought you might mean "いそがしい, isogashii" I thought the following was interesting: "Derived from the verb 急ぐ (isogu, “to hurry, to rush, to hasten”), using the adjectivizing suffix しい (-shii, “having that kind of quality”)."
Why was そのレストラン translated to "The restaurant"? Wouldn't that correspond to レストランが。。。?
Out of curiosity, since the adjective is not connected to the noun, and the sentence is in the past tense, why are we conjugating the です copula and not the adjective? Why isn't it something like, そのレストランはにぎやかったです? (This is probably horribly mangled, but I'm just using it as an example of what it might look like.)
Japanese mainly has two kinds of adjectives: -i adjectives and -na adjectives.
-i adjectives use an い to complement a noun (in their present affirmative form), and are conjugated as we were taught. For example, the color red, 赤 (あか), becomes 赤い (あかい) when used as an adjective, and can become 赤かった, 赤くない, etc.
-na adjectives use a な when they are next to the noun which they modify, and are not conjugated (the conjugation is done in the verb). For example, using しずか (which means "quiet"), "a quiet person" is しずかな人 (the な is connecting the adjective and the noun), "I am quiet" is 私はしずかです (here the adjective is not next to a noun, so we don't need the な), and "I was quiet" is 私はしずかでした .
So only -i adjectives can be conjugated that way. Remember that in order to conjugate them, you have to transform the い into something else. Most -na adjectives don't end in い (although some do), so you don't even have a part that you can modify.
Finally, as you were probably expecting already, にぎやか is a -na adjective, so that is why it is not conjugated. Just to reinforce a little the concept: "A lively restaurant" is にぎやかなレストラン
Hope that helped!
The way I was taught was that -na adjective + でした is the way -na adjectives are conjugated for positive past tense. And -い adjectives are conjugated dropping the い and adding かったです.
This is more or less what you just said and either way it comes out the same. However, I would find it confusing to think of the -na adjectives as not being conjugated at all. It's easier for me to think of the -na adjectives and -i adjectives as being conjugated differently. It just so happens that the -na adjectives are conjugated the same way as です.