Fingers crossed the pup has learned his lesson.

I'm not sure what you mean by "explain the sentence in terms of grammar", but I can tell you that it's another way of saying "We hope that this little dog has learnt a lesson from this experience (and therefore won't repeat it)". Did you look up "fingers crossed" anywhere?

Fingers crossed​ is a sentence adverb. It means the writer hopes the puppy has learned something.

I thought it was subject so didn't feel the need to look it up

capcap23 said:

I thought it was the subject so I didn't feel the need to look it up.

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It doesn't matter whether something is the subject or not - if you don't know what it means, you need to look it up.

emsr2d2 said:

if you don't know what it means, you need to look it up.

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There is no way I can either tell or find the idiom. I know the words as they are.

I don't know what post #6 means. Can you explain, using other words, what you thought "Fingers crossed" meant?

emsr2d2 said:

I don't know what post #6 means. Can you explain, using other words, what you thought "Fingers crossed" meant?

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Fingers draw a line on dog has learned his lesson. This is what I thought before I learnt it is adverb.

capcap23 said:

I thought it was subject so didn't feel the need to look it up

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There should be a comma after it IMO to separate it from the rest of the sentence.

Hmm, I debated the comma aspect. I sometimes use one with "Fingers crossed" and sometimes I don't. It depends on what comes next.

Fingers crossed we get it done by midnight.
Fingers crossed, we'll get it done by midnight.

Given that the present tense is used in post 1, I wouldn't use the comma. I would use it if it said "Fingers crossed, the pup will have learnt his lesson".

Fingers crossed, the pup has learned his lesson.
=
Hopefully, the pup has learned his lesson.