A veces dejo el móvil en silencio y luego encuentro un mensaje en el contestador.

Because Spanish often uses the simple present to talk about habitual actions. In this sentence, the speaker is describing something that happens from time to time:

• A veces dejo... = Sometimes I leave...
• luego encuentro... = then I find...

In English, we also use the present simple for habits: Sometimes I leave my phone on silent and then I find a message.

Spanish usually drops subject pronouns when the verb ending already shows who the subject is.

• dejo = I leave
• encuentro = I find

So yo is not necessary here. You could say Yo a veces dejo..., but it would usually sound more emphatic, as if you were stressing I.

In Spanish, it is very common to use the definite article instead of a possessive adjective when the owner is obvious from the context.

So:

• dejo el móvil en silencio
  literally = I leave the mobile on silent
• natural English = I leave my phone on silent

Since the speaker is talking about their own phone, Spanish does not need mi.

A veces means sometimes.

It is one of the most common ways to express occasional frequency in Spanish. Algunas veces is also possible and means roughly the same thing, but a veces is shorter and more common in everyday speech.

Examples:

• A veces llego tarde. = Sometimes I arrive late.
• A veces no contesta. = Sometimes he/she doesn’t answer.

Yes, literally it is something like leave the phone in silence, but idiomatically it means:

• put the phone on silent
• leave the phone on silent mode

The verb dejar often means to leave something in a certain state:

• dejar la puerta abierta = leave the door open
• dejar el móvil en silencio = leave the phone on silent

So here en silencio describes the phone’s mode or state.

Because en silencio is a fixed and natural way to express the idea of being on silent or in silent mode.

• silencioso means silent/quiet as an adjective, but it would describe something as inherently quiet, not usually as a device setting.
• For phones, Spanish normally says poner/dejar el móvil en silencio.

You may also hear:

• en modo silencio
• en vibración / en modo vibración

But en silencio is very natural.

Yes, here luego means then or afterwards, and in this sentence it works very much like después.

So these are both possible:

• ...y luego encuentro un mensaje...
• ...y después encuentro un mensaje...

Luego is very common in everyday Spanish. In Spain, it often sounds slightly more conversational and natural in this kind of sequence.

Yes. In Spanish, encontrar is a very natural verb for discovering that a message is there.

So:

• encuentro un mensaje en el contestador
  literally = I find a message on the answering machine
• natural English = I find a message / I discover there’s a message on voicemail

Spanish often uses encontrar where English might prefer see, notice, or find that there is.

In Spain, contestador usually refers to an answering machine or voicemail service.

Depending on context, it can mean:

• a physical answering machine
• your phone’s voicemail

A fuller form is contestador automático, but people often just say contestador.

In modern everyday speech, especially with mobile phones, many speakers might also say:

• buzón de voz = voicemail box / voicemail

So un mensaje en el contestador is perfectly understandable as a voicemail message.

Because Spanish treats the message as being in/on the answering machine or voicemail system.

• en el contestador = on the answering machine / in voicemail

Other prepositions would change the meaning:

• del contestador = from the answering machine
• al contestador = to the answering machine

So en is the correct choice when the message is located there.

Yes, móvil is the standard word in Spain for mobile phone / cell phone.

In much of Latin America, learners are more likely to hear:

• celular
• sometimes móvil, depending on the country

So this sentence sounds clearly natural for Spain Spanish.

Yes, several variations would sound natural in Spain. For example:

• A veces dejo el móvil en silencio y luego veo que tengo un mensaje en el contestador.
• A veces pongo el móvil en silencio y luego encuentro un mensaje en el contestador.
• A veces dejo el móvil en silencio y después tengo un mensaje en el buzón de voz.

The original sentence is already completely natural, but Spanish offers a few different ways to express the same idea depending on style and context.

The word order is natural and standard, but Spanish is somewhat flexible.

The original order works well because it presents:

1. the time expression: A veces
2. the first action: dejo el móvil en silencio
3. the next action: y luego encuentro un mensaje en el contestador

You could move some parts for emphasis, for example:

• A veces, luego encuentro un mensaje en el contestador porque dejo el móvil en silencio.

But that changes the rhythm and focus. The original version is the most straightforward and natural for everyday use.