sich bemühen
to strive
To make an effort or try hard in order to achieve a specific goal.
Sie bemüht sich sehr, ihre Deutschkenntnisse zu verbessern.
She is trying very hard to improve her German skills.

to strive / to trouble
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to strive
To make an effort or try hard in order to achieve a specific goal.
Sie bemüht sich sehr, ihre Deutschkenntnisse zu verbessern.
She is trying very hard to improve her German skills.

'Bemühen' (to strive / to trouble) is related to the noun 'die Mühe' (effort). The inseparable prefix 'be-' turns it into a verb that describes an action.
Er bemüht sich, pünktlich zu sein.
He is trying to be on time.
The meaning depends on the object: Is it used with a reflexive pronoun ('sich bemühen' — to make an effort), a person ('jemanden bemühen' — to trouble someone), or a thing ('einen Vergleich bemühen' — to invoke a comparison)?
Ich bemühe mich um eine schnelle Antwort.
I am making an effort to provide a quick answer.
In this reflexive meaning, there are two typical constructions: 'sich bemühen um + Akkusativ' (to make an effort to get/obtain something) or 'sich bemühen, etwas zu tun' (to try to do something; infinitive with zu).
Der Student bemüht sich um ein Stipendium.
The student is trying to get a scholarship.
The expression 'jemanden bemühen' (to trouble someone) is a very polite and formal way to ask someone for help. It expresses that you are aware of the inconvenience caused.
Dafür müssen Sie nicht extra den Chef bemühen, das kann ich für Sie erledigen.
You don't need to trouble the boss especially for that; I can take care of it for you.
This meaning is often used in argumentative or intellectual contexts. Typical objects are: einen Vergleich bemühen (to invoke a comparison), ein Beispiel bemühen (to invoke/use an example), eine Statistik bemühen (to cite/use a statistic).
Um seinen Standpunkt zu stärken, bemühte er eine aktuelle Studie.
To strengthen his point of view, he cited a recent study.
to try / taste
'versuchen' and 'sich bemühen' both describe wanting to achieve a goal even though success is not certain. 'versuchen' emphasizes the action more: you try something out or make an attempt. 'sich bemühen' emphasizes the effort and good intentions. It often sounds a bit more serious or polite. Learners may confuse the words because both have to do with a goal and possible success. If you want to say that someone is trying something, use 'versuchen'. If you want to show the effort and commitment, 'sich bemühen' fits better.

Das Kind bemüht sich beim Lesen, auch wenn manche Wörter noch schwer sind.
The child makes an effort when reading, even though some words are still difficult.
Im Kurs habe ich versucht, die neue Grammatik ohne Hilfe zu erklären.
In the course, I tried to explain the new grammar without help.