The MCL or medial collateral ligament is located in the knee on the side closest to the other knee. It helps stabilize the knee joint and prevent injury when force is directed through that side of the knee. An isolated injury to this ligament could very well heal with rest and activity modification.
Not all ligaments have a good blood supply in order to create a healing response. But the MCL does have more of a healing capacity than most ligaments. If there are no other injuries to the meniscus, cruciate ligaments, or other soft tissues, you may expect a good recovery from this type of injury.
If the knee is stable (not giving out from under you), then a brace isn’t usually needed. And since you have seen a physician, it sounds like your knee was examined before making the recommendations to give it some time.
With a history of injury and reinjury, if you experience further problems, don’t wait to get back in to see the physician for a follow-up visit. A supportive splint and exercises may be advised at that point.
Surgery is rarely done for an acutely injured MCL unless it is part of an injury that has damaged multiple knee ligaments and/or soft tissues. When joint laxity threatens the integrity of the joint, then surgery to repair or reconstruct the soft tissues (including the MCL) may be necessary.