Ma mère et ma tante s’écrivent l’une à l’autre le soir.

Because French is showing that the action goes between the people in the subject.

The base verb is écrire = to write.
But écrire à quelqu’un means to write to someone.

So in Ma mère et ma tante s’écrivent, the s’ stands for to each other / to themselves, depending on context. Here, with two people, it means to each other.

So the idea is not just they write, but they write to each other.

Here it is reciprocal, not truly reflexive.

• Reflexive: the subject does something to itself
• Reciprocal: the people in the subject do the action to one another

So in this sentence:

• Ma mère et ma tante s’écrivent = my mother and my aunt write to each other

Because there are two people, the natural reading is reciprocal.

Because French often uses both forms together for clarity or emphasis.

So:

• Ma mère et ma tante s’écrivent le soir
• Ma mère et ma tante s’écrivent l’une à l’autre le soir

Both are possible, but the second one is more explicit: it clearly says one to the other.

This is very common with reciprocal actions in French.

Because the two people are feminine:

• ma mère
• ma tante

So French uses the feminine form:

• l’une à l’autre

If the two people were masculine, or if the group were mixed, French would normally use:

• l’un à l’autre

Because the verb écrire takes à before the person receiving the writing.

You write:

• écrire à quelqu’un = to write to someone

So the reciprocal expression follows that pattern:

• l’une à l’autre = to one another

This depends on the verb. For example:

• Ils se regardent l’un l’autre
  because regarder quelqu’un has no à
• Ils se parlent l’un à l’autre
  because parler à quelqu’un uses à

So s’écrivent l’une à l’autre matches the grammar of écrire à quelqu’un.

Because in French, object pronouns normally go before the conjugated verb.

So French says:

• elles s’écrivent

not something like:

• elles écrivent-se

Also, se becomes s’ before a vowel sound. Since écrivent begins with é-, you get:

• se écrivent → s’écrivent

This is called elision.

Le soir usually means in the evening or in the evenings, often with a habitual sense.

In French, the definite article is often used in time expressions like this:

• le matin = in the morning
• l’après-midi = in the afternoon
• le soir = in the evening

So here, le soir suggests something like in the evenings or at night as a regular time when they write.

If you wanted this evening, you would say:

• ce soir

It is in the present tense.

But the French present tense can mean different things depending on context. Here, with le soir, it most naturally suggests a habit or regular action:

• they write to each other in the evenings

So this sentence does not strongly suggest that they are writing at this exact moment.

The verb part would work the same way, but the reciprocal expression would usually change form.

• Two feminine people: l’une à l’autre
• Two masculine people: l’un à l’autre
• One masculine and one feminine: usually l’un à l’autre as well

So for example:

• Mon père et mon oncle s’écrivent l’un à l’autre le soir.
• Ma mère et mon oncle s’écrivent l’un à l’autre le soir.

The verb s’écrivent stays plural because there are two people in the subject.