Learn Russian: Без десяти шесть он смотрел на табло и начинал нервничать. - At ten to six, he was looking at the display board and starting to get nervous.

This is the standard Russian formula for minutes to the hour. The preposition без (without) always forces the number following it into the genitive case, so 'ten' becomes десяти. The upcoming hour that you are approaching is just stated in the nominative case, so it remains шесть.

Do we need a preposition like в to say 'at ten to six'?

No, you do not need to add в (at) when using the без time construction. The phrase без десяти шесть handles both 'it is ten to six' and 'at ten to six'. The rest of the sentence makes it clear that this is the time an action was happening.

Why doesn't the ending of табло change after the preposition на?

The word табло (display board) is a borrowed foreign word. In Russian, most borrowed nouns ending in -о, -е, or -и (like кино, метро, or кафе) are indeclinable. It stays exactly the same in all cases, even though here it is functioning as the accusative object after на.

Why does the sentence use the imperfective past verbs смотрел and начинал instead of their perfective equivalents?