L’électricienne laisse son numéro au serrurier, au cas où ils devraient revenir le même jour.

Because électricienne starts with a vowel sound, la becomes l’ in front of it. This is called elision.

• la secrétaire
• l’électricienne

So l’électricienne simply means the electrician in the feminine form.

French job titles often change form depending on the person’s gender.

• un électricien = a male electrician
• une électricienne = a female electrician
• un serrurier = a male locksmith
• une serrurière = a female locksmith

In this sentence, the electrician is female, so French uses l’électricienne.

Because French possessive adjectives agree with the thing possessed, not with the owner.

Here, the possessed thing is numéro, which is a masculine noun:

• un numéro → son numéro
• une voiture → sa voiture

So son numéro means her number, even though son is the masculine possessive form.

It is a very common expression meaning to leave one’s number, usually a phone number.

So:

• L’électricienne laisse son numéro au serrurier
  = The electrician leaves her number with / gives her number to the locksmith.

It does not mean she is physically abandoning a written number somewhere; it is just the normal way to say this.

Because au is the contraction of à + le.

• à le serrurier → au serrurier

Here à marks the person receiving something:

• donner quelque chose à quelqu’un
• laisser quelque chose à quelqu’un

So au serrurier means to the locksmith or with the locksmith, depending on context.

Because au cas où introduces an extra idea like in case. In writing, it is often separated by a comma, especially when it comes after the main clause.

• L’électricienne laisse son numéro au serrurier, au cas où...

This is similar to English punctuation in sentences like:

• She leaves her number with the locksmith, in case...

The comma is natural here, though punctuation can vary a bit depending on style.

Au cas où means in case.

It introduces a possible situation:

• au cas où il pleuvrait = in case it rained / in case it should rain
• au cas où ils devraient revenir = in case they had to come back

It is a fixed expression, and one important thing about it is that standard French normally uses the conditional after it.

Because after au cas où, standard French normally uses the conditional.

So:

• au cas où ils devraient revenir = in case they should have to come back

Here devraient is the conditional present of devoir.

This does not mean the same as a plain future or present. It expresses a possibility, not a fact.

Very roughly:

• ils doivent revenir = they have to come back / they must come back
• ils devaient revenir = they were supposed to come back / they had to come back
• ils devraient revenir = they might have to come back / should have to come back

After au cas où, the conditional is the standard choice.

Not exactly in the everyday advice sense of should.

In English, should often means ought to, but here French devraient after au cas où means something more like:

• if they happened to need to come back
• in case they had to come back
• in case they should need to come back

So it expresses a possible future necessity, not advice.

Ils most naturally refers to the people already mentioned, especially if they might both need to return.

In this sentence, it likely means the electrician and the locksmith.

French uses ils for a mixed-gender group, even if one person is female and one is male.

• one woman + one man → ils
• two women → elles

So even though the electrician is female, ils is correct because the group includes a man.

In French grammar, a mixed group takes the masculine plural.

Since the sentence has:

• l’électricienne = female
• le serrurier = male

the pronoun for both together is ils, not elles.

Revenir means to come back or to return.

It is very natural here because the idea is that they may need to come back again to the same place. French often uses revenir in this kind of situation.

You may also know retourner, which can also mean to return/go back, but revenir is especially common for come back.

Le même jour means the same day.

• même = same
• le même jour = the same day

So the sentence means they might need to return later that very day, not on another day.

Because the phrase points to a specific day already understood from the context: that same day.

French usually says:

• le même jour = the same day
• la même chose = the same thing
• la même personne = the same person

Using un même jour would not sound natural here.

Yes. In English, both can work depending on the situation.

• She leaves her number to the locksmith sounds a bit unnatural in English.
• She leaves her number with the locksmith is more idiomatic.
• She gives her number to the locksmith is also very natural.

In French, laisser son numéro à quelqu’un is the normal structure.

It is perfectly normal, standard French. Nothing in it is especially formal or literary.

A native speaker could easily say or write this in everyday life, especially in a work or service context. The only point that learners often notice as slightly more grammar-sensitive is the use of the conditional after au cas où, which is standard French.