You will also find more information in the Tips and Notes here:
https://www.duolingo.com/skill/fr/Verbs%3A-Present-3

i.e.
nous nous levons : we get up
vous vous réveillez : you wake up

Also remember about elision , that is a factor for me/te/se ref: wikia
Thus you get:
m'asseoir : to sit (down) - I sit down
t'appeler : to be called - you (familiar) are being called.
s'habiller : to get dressed - he/she gets dressed.

related subjects :
- Reflex verbs : where the subject acts on itself.
- Reciprocal verbs : where the subjects act on one another.
- Idiomatic pronominal verbs : the reflexive pronoun changes the meaning of the verb.
- Essentially pronominal verbs : where the verb can only be used in a pronominal way.

A pronominal verb is a verb that is accompanied by a reflexive pronoun. Pronominal verbs fall into three major classes based on their meaning: reflexive, idiomatic, and reciprocal. ... The verb is conjugated normally (here an -er verb) with addition of the reflexive pronouns me, te, se, nous, vous, se. ref. : www.laits.utexas.edu

So , in short , if you are talking about yourself, you should use reflexive verbs.
In this case m’améliorer.

If you do not use reflexive verbs, then it is assumed you are talking about something other than yourself. In this case, you are hoping something or someone outside of you will improve.

It probably works best to explain, if I use a different verb.

If you said :
Je lave. in French it would be assumed you were washing someone or something else.

To wash yourself you need to say :
Je me lave.

As always, if I have misunderstood something, I trust someone will correct me.

In French, when one does something to oneself, you use a reflex verb.

So in your example, it is correct to use a reflex verb. :
J’étudie le français chaque jour, donc j’espère m’améliorer. :
I study French every day, so I hope to improve myself.

Whenever a verb (doing word) , is about where you do something to yourself, it is correct to use a reflex verb.

There are though some tricks about reflexive verbs that might interest you further.

I will come back with some more information, or perhaps someone else could chime in.

Also I would suggest checking out:
https://www.lawlessfrench.com/grammar/reflexive-verbs/
https://www.lawlessfrench.com/grammar/reflexive-pronouns/
https://www.lawlessfrench.com/grammar/reciprocal-verbs/
https://www.lawlessfrench.com/grammar/agreement-with-pronominal-verbs/

I don't agree that you should use a reflexive verb the way your sentence is constructed. You're not improving 'yourself', you're improving 'it', your French. Thus, "J'étudie le français ........ j'espère l'améliorer."

Thanks, everyone. I appreciate it.