Tus mensajes de ánimo me animan a seguir practicando español cada día.

They come from the same word family, but they are doing different jobs here:

• ánimo is a noun in mensajes de ánimo. It means something like encouragement, support, or motivation.
• animan is a verb form from animar. Here it means encourage.

So:

• mensajes de ánimo = messages of encouragement
• me animan = they encourage me

This repetition is completely natural in Spanish. In English we might avoid repeating encourage / encouragement, but Spanish often allows it if it sounds clear and natural.

Mensajes de ánimo is a very common expression meaning:

• encouraging messages
• messages of support
• messages that lift someone’s spirits

The structure is:

• mensajes = messages
• de ánimo = of encouragement / of support

Here ánimo is not literally mood in the everyday English sense. In this phrase, it is more like encouragement, heart, or moral support.

For example:

• Gracias por tus palabras de ánimo. = Thanks for your encouraging words.
• Necesito un poco de ánimo. = I need a bit of encouragement.

Because mensajes is plural.

Spanish possessives agree in number with the thing possessed:

• tu mensaje = your message
• tus mensajes = your messages

So since there is more than one message, Spanish uses tus.

Because the subject is plural: Tus mensajes.

The verb must agree with its subject:

• Tu mensaje me anima = Your message encourages me
• Tus mensajes me animan = Your messages encourage me

So animan is the third person plural form, matching mensajes.

Here me means me, as the person being encouraged.

In this sentence, animar works like this:

• animar a alguien a hacer algo = to encourage someone to do something

Example with a noun:

• Tus mensajes animan a Juan a seguir.

When the person is replaced by a pronoun, Spanish uses:

• Tus mensajes me animan a seguir.

So me is the object pronoun for me.

A useful way to think about the whole pattern is:

• X anima a alguien a hacer algo
• X encourages someone to do something

Because animar is commonly followed by the pattern:

• animar a alguien a + infinitive

So:

• me animan = they encourage me
• a seguir practicando = to keep practising

The second a is part of the verb pattern, not something optional.

More examples:

• Me anima a estudiar. = It encourages me to study.
• Nos animaron a participar. = They encouraged us to take part.

So in your sentence:

• Tus mensajes de ánimo me animan a seguir practicando español cada día.

the structure is perfectly regular.

Because after seguir, Spanish normally uses a gerund when it means to keep doing something.

So:

• seguir practicando = to keep practising
• seguir estudiando = to keep studying
• seguir hablando = to keep speaking

This is one of the most common uses of seguir + gerundio.

Compare:

• Sigo practicando español. = I keep practising Spanish.
• Quiero seguir practicando español. = I want to keep practising Spanish.

Using seguir practicar would sound wrong here.

No. With this meaning, the normal structure is seguir + gerund:

• seguir practicando

Not:

• seguir a practicar
• seguir de practicar

Those do not fit this grammar pattern.

So the key chunk to remember is:

• seguir + gerundio = to keep doing something

Because here español refers to the language in a general sense after practicar.

In Spanish, language names often appear without an article after verbs like:

• hablar
• estudiar
• aprender
• practicar

So:

• practicar español
• aprender español
• hablo español

Using el español is possible in some contexts, but here it would sound less natural.

So:

• practicando español = practising Spanish

is the normal choice.

Cada día means every day or each day.

In this sentence, it is very close in meaning to todos los días:

• practicando español cada día
• practicando español todos los días

Both are natural.

A small difference in feel:

• cada día can sound a little more like day by day / each day
• todos los días can sound a little more straightforwardly habitual: every single day

But in many everyday sentences, they are basically interchangeable.

Because ánimo is stressed on the first syllable:

• Á-ni-mo

The accent mark shows that stress clearly.

Also, the accent helps distinguish it from other forms such as:

• animo = I encourage / I cheer up
• ánimo = encouragement / spirit / courage

So:

• Te animo a seguir. = I encourage you to continue.
• Necesito ánimo. = I need encouragement.

Yes, it sounds natural and correct in Spanish from Spain.

It has a warm, personal tone and would fit well in contexts like:

• replying to a friend
• thanking someone online
• talking about language learning motivation

A native speaker might also say similar versions such as:

• Tus mensajes de ánimo me ayudan a seguir practicando español cada día.
• Tus mensajes me animan a seguir practicando español todos los días.

But your original sentence is absolutely natural.

Not exactly. In this sentence, animan is better understood as:

• encourage me
• motivate me

Cheer me up usually means making someone feel happier when they are sad.

But here the idea is more:

• your encouraging messages motivate me to continue practising

So encourage is the best match in this context.

A helpful breakdown is:

• Tus mensajes de ánimo = subject
• me animan = verb + object pronoun
• a seguir practicando español = infinitive phrase showing what I am encouraged to do
• cada día = time expression

So the sentence follows this pattern:

• [Subject] + animar + a alguien + a + infinitive

More abstractly:

• Your encouraging messages encourage me to keep practising Spanish every day.

Even though the English wording may feel repetitive, the Spanish structure is very normal.