Мне стоит убрать телефон, чтобы не отвлекаться на чат.

It’s worth putting my phone away so I don’t get distracted by the chat.

Breakdown of Мне стоит убрать телефон, чтобы не отвлекаться на чат.

• на: by
• телефон: the phone
• не: not
• мне: me

Questions & Answers about Мне стоит убрать телефон, чтобы не отвлекаться на чат.

Why does the sentence start with Мне, and why is it in the dative case?

Мне is the dative form of я (to me / for me). In Russian, feelings, necessity, advice, and “it would be good for me to…” ideas are often expressed with a dative experiencer rather than a nominative subject.
So Мне стоит… literally means something like It is worth (it) for me to…, i.e. I should / It would be a good idea for me to….

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What does стоит mean here, and is it related to стоить (“to cost”)?

Yes—this is the verb стоить. Besides to cost, it also has the meaning to be worth (doing).
Pattern: (кому) стоит + infinitive = someone should / it’s worth for someone to…
So Мне стоит убрать… = I should put away… / It’s worth it for me to put away….

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Is Мне стоит… strong like I must, or softer like I should?

It’s usually softer than “must”. It sounds like good advice or a personal conclusion: I should / It’d be a good idea / I’d better.
If you want stronger obligation, you’d more often use мне надо / мне нужно / мне придется.

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