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Hello, this is Dr. Jess. I am very sorry to hear about your kitten! Unfortunately, it sounds like there is a very good possibility this kitty has a fractured leg, given the history of being stepped on and the swelling. Usually if we just have a soft tissue injury like a sprain we will not see the extensive swelling you are describing.
Unfortunately "at home care" alone is not recommended in these cases for several reasons. It is possible that the fracture may be such that only splinting is required, but we will not know that until we x-ray the leg. It is possible the fracture is such that a splint is not going to be adequate and no amount of splinting the bone will allow it to heal. Another reason is that it is definitely recommended for a veterinary professional to place a splint on your kitty, as it is quite common for owners to accidentally end up causing more harm than good when trying to bandage or splint their pet's limbs at home on their own. The leg can end up with pressure sores, incorrect positioning leading to healing improperly, or a bandage causing decreased circulation in the leg. The final reason veterinary care is recommended is because this kitty is likely in need of pain medication as well, and there is no safe at-home/over the counter pain medication to give to cats.
For all of these reasons, your kitten truly needs to be seen in person by a vet. Probably not what you want to hear, but I don't want to lie to you and tell you that this is something you can fix at home! In the meantime, keep him confined to a small crate or box to limit his movement as much as possible.
Please let me know if you have any additional questions. If you do not, please take a moment to CLICK AND RATE our conversation so we can know if you were satisfied with your service and I can get credit for helping to answer your question. Thank you and best of luck to your kitty!