This is a bit of a mischaracterisation of what happened. No-one forced the Senate as a body to flee Rome; Pompey and many of the senators left, but several stayed (according to Cassius Dio), and they even met as a body just outside Rome shortly after Caesar entered.

However, the major effect of much of the Senate leaving was that Caesar gained - or more accurately, gave himself - unfettered access to the public treasury in Rome, again as referenced in Cassius Dio, which could finance his campaigns against Pompey et al.

It is hard to divorce the evacuation of Pompey and the senators from the rest of the context of 49 BC and the civil war which it was part of, which certainly had a significant effect; Dio again tells us that financial uncertainties due to the civil war caused issues with loaning money, for instance.