Why do the definitions of non-deterministic Turing machines look strange?
Basically, a NDTM on input $x$ may have more than one computation branches. If at least one of these branches accepts the input then we say the NDTM accepts the input $x$, otherwise it rejects.

You can consider a NDTM as a finite number computation branches such that branch on input$x$ may halt with accept or reject, or run forever. If at least one branch halts with accept then the NDTM accepts that input no matter other branches halts or run forever. If no branch accepts (halts with in ACCEPT state) then we say the NDTM does not accept the input. And if every branch halts and rejects then we say that NDTM rejects the input.

Do not try look for a real life model of a NDTM. It is just a conceptual computation model which allows to define a complexity class known as $\mathrm{\mathbf{NP}}$ class.

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