So, this sentence pretty much means, it's good, heal quickly. However, I don't understand how から contributes to the sentence. Can anyone explain?
良い often functions like "ok" / "fine" / "alright" in English in that its base meaning is positive but it can be used negatively, sarcastically, or to otherwise express a lack of need, and so in some fixed expressions it is predominantly negative.
Look at these example conversations and their very loose translations:
(A: Shall I prep some copies of these documents? B: That's alright.)
(A: Could I make some copies of these documents? B: That's alright.)
You can see that even in English intonation and context can give the exact same reply seemingly opposite yes and no usages.
In daily life, いいです or 大丈夫です can often functionally mean 要りません in a roundabout way , similar to how in English "okay" can mean "yes" while confusingly "I'm okay" is a way to say "I don't need that", or "no", while "that's okay" can mean either depending on the context and tone. I feel the 'negative' sense is perhaps used even more commonly than in English, for example レジ袋はよろしいですか at a store is a common expression (saying はい results in no bag!).
The other answer provides the excellent translation of "alright already" for いいから , but there are other phrases like もういい and よくも that also carry this nuance.
As oldergod's answer says, いいから is an idiom and is not a combination of いい and から. The meaning is closest to the English idiom "alright already". For example, if someone is in the hospital but all they're worried about is all the work they have to do and how their boss is going to yell at them when they get back, you might say to them: "alright already, you'll deal with that later, for now just get better".
The idiom tends to get used when the listener is overly worried with something irrelevant or trivial, and particularly when they won't shift their focus to what's actually important (in the opinion of the speaker).