In general, the 1850 law strengthened the ability of slaveholders to claim their property (which is what slaves were considered) and greatly weakened the slave's (or suspected slave's) ability to defend him/herself. Government officials were required to help slaveholders in their pursuit, and could in turn deputize any citizen to help. Perhaps most notably, suspects could be returned (or even brought) to slavery very easily:

...a claimant could bring an alleged fugitive before a federal commissioner (a new office created by the law) to prove ownership by an affidavit from a slave-state court or by the testimony of white witnesses. If the commissioner decided against the claimant he would receive a fee of five dollars; in in favor, ten dollars.

(Battle Cry of Freedom)

In other words, a slaveholder could bring a black man or woman before the commissioner, have a 'witness' affirm that s/he was a slave, and leave with them. This opened tremendous possibilities for abuse, to the point that there was a mass migration out of not only the South, but much of the North as well (Canada and England were popular destinations for free black people who didn't want to worry about the long arm of the law).