Paramedics wear gloves to protect themselves from diseases, viruses, and bacteria (as well as dirty bodily fluids, such as urine, fecal matter, vomit and many more). Paramedics also wear gloves so that the patient, who may be immunocompromised (their imune system has been lowered due to cancer treatment or disease) are not made sick or unwell because of the paramedic’s diseases or viruses. This is particularly important when a paramedic cannulates, gives injections, or treats open wounds in which there is a risk of transfering bacteria and viruses through the opening in the skin.
The reason paramedics wear blud gloves, instead of white gloves, is blue gloves are generally made out of Nitrile (which has a very very low allergenic property), where as white gloves are made out of latex, which many people are commonly allergic to. Because paramedics do not know if the person that they are attending are going to be allergic to latex or not, they treat everyone while wearing nitrile gloves.
This is the same for police officers who may wear blue nitrile gloves if they are helping people.