Mi amiga se siente más segura de sí misma cuando habla en público.

Because sentirse means to feel in the sense of to feel a certain way.

• sentir = to feel, to sense, to regret
  
  • Siente frío = She feels cold
  • Siento alegría = I feel joy
• sentirse = to feel oneself, to feel in a certain state
  
  • Se siente segura = She feels confident/safe

So in this sentence, se siente is the natural verb because it describes her emotional state.

Because segura agrees with mi amiga, which is feminine singular.

In Spanish, adjectives usually match the noun they describe in:

• gender: masculine/feminine
• number: singular/plural

So:

• mi amigo se siente más seguro
• mi amiga se siente más segura

Since amiga is feminine singular, segura must also be feminine singular.

Here, de sí misma means in herself or about herself.

The expression seguro/a de sí mismo/a is a very common way to say self-confident or confident in oneself.

So:

• segura de sí misma = confident in herself / self-confident

Without de sí misma, segura could just mean sure, safe, or certain, depending on context.
Adding de sí misma makes it clear that this is about self-confidence.

Because sí and si are different words.

• sí = himself / herself / itself / themselves after a preposition; also yes
• si = if

In this sentence, sí refers back to the subject mi amiga after the preposition de:

• de sí misma = of herself / in herself

So the accent is important here.

Misma adds emphasis and is very common in this structure.

• sí already refers back to the subject
• misma strengthens it: herself

So:

• de sí misma = in herself / in herself herself, in the emphatic sense

This is a standard expression:

• seguro de sí mismo
• segura de sí misma

Misma also agrees with amiga, so it is feminine singular.

Because Spanish can use más + adjective without stating the second part of the comparison when it is understood from context.

So más segura can mean:

• more confident than before
• more confident than usual
• more confident in that situation

The comparison is implicit.

If you wanted to make it explicit, you could say things like:

• más segura que antes = more confident than before
• más segura que en otras situaciones = more confident than in other situations

But the sentence sounds completely natural without that extra information.

Because this sentence describes a habitual or general situation.

• cuando + present indicative = when/whenever something usually happens
• cuando + subjunctive = when referring to a future event that has not happened yet

Here the meaning is general:

• cuando habla en público = when she speaks in public / whenever she speaks in public

If you were talking about a future occasion, you might say:

• cuando hable en público mañana = when she speaks in public tomorrow

So the indicative habla is correct here because this is not a one-time future event.

Because the subject of habla is still mi amiga.

The sentence is about one person:

• Mi amiga = my friend
• se siente = she feels
• habla = she speaks

Spanish often does not repeat the subject if it is already clear. So habla is automatically understood as ella habla.

If it were plural, then you would use hablan:

• Mis amigas se sienten más seguras de sí mismas cuando hablan en público.

Because Spanish usually leaves out subject pronouns when they are not needed.

The verb ending already tells you the person:

• habla = he/she/it speaks

Since the subject mi amiga was already mentioned, repeating ella would usually be unnecessary.

• Natural: Mi amiga se siente más segura de sí misma cuando habla en público.
• Also possible, but more emphatic: Mi amiga se siente más segura de sí misma cuando ella habla en público.

Spanish generally prefers the version without the pronoun unless there is a reason to stress it.

En público means in public, before other people, or in front of an audience.

It does not have to mean being out in a public street or place. In this context, it usually means something like:

• speaking before a group
• public speaking
• talking in front of other people

So habla en público is the normal expression for speaks in public.

Yes, but the nuance is a little different.

• se siente más segura de sí misma focuses on how she feels
• está más segura de sí misma focuses more on her current state

In many contexts, both are possible. But se siente is especially natural when talking about confidence as an inner feeling.

So this sentence sounds very natural because it emphasizes her personal experience:
she feels more confident when speaking in public.