Usually this is solved by just changing the door. Its not a woodworking beginner's fix.

Not necessarily the best solution...

Many years ago I had an office break-in by punching out a vent in the bottom of a door. Landlord was useless, so I got a couple of pieces of plywood, put one on each side of the door to cover the hole and put some big screws through plywood/door/plywood. Needless to say, we moved out of the office not long after.

Years later, all the landlord ever did was to paint the plywood to match the rest of the door!

But if I actually owned the place, I would have replaced the door. Which is what I recommend you do.

The long answer. For the TL;DR, skip to the bottom

A repair isn't difficult per se, but it will take time, energy and tools.

That bottom rail of the door is a separate piece of wood glued to the two vertical stiles on either side. It is likely a reasonably standard "cope and stick" cut on the ends. You will need a router and a "cope and stick" bit set to make the cut.

• Take the door off the hinges and lay it down on some saw horses.
• Put up a temporary door, because it's going to be off the hinges for a couple of days
• Using a saw, cut that bottom rail near each end to remove the bulk of the wood
• Measure the vertical size of the piece you just cut out, and the thickness of it. Write this down, you'll need it later.
• Remove the little pieces of rail that are left attached to the stiles. You'll be breaking a glue joint, and often, the glue is stronger than the wood. Heat will often soften the glue allowing it to be removed, but there's no guarantee. It will depend on the type of glue used. Asking about glue separation on the Woodworking.SE when you get stuck would be a good idea.
• Pull the cut up panel out of the door. This should be easy - it should not be glued in. Wood expands & contracts, so the panel is left floating so the door doesn't tear itself apart.
• Buy a piece of plywood the same thickness as this original panel and cut it to the same size as the panel. Set the new panel aside, chuck the old one into the burn pile.
• Find a "cope and stick" bit set for your router that matches the profile of your door. You may want to take the big piece of rail you cut out to your local woodworker's supply place (Rockler and Woodcrafter's are two big chain stores in the US, as examples of what I'm talking about. If you've got a similar non-chain local store, I'd encourage supporting them) to have them help you identify what set will work for you.
• Cut a replacement rail oversize for the dimensions you need.
• Cut the cope along what will be the top (facing the panel)
• Cut the "stick" along one edge.
• Carefully measure the length and cut the "stick" along the other edge.
• Slot the new plywood panel into place
• Liberally apply glue to the rails and to the stick cuts on the new rail
• Slide it into place in the door, making sure you've got good glue contact with the stiles, but not with the panel
• Clamp the stiles across the rail to hold it tight while it dries.
  • Tighten the clamps firmly
  • No, more than that
  • No, really, even tighter!
  • Now, add a couple more clamps
  • Yes, more than that
  • As a matter of fact, if you can have clamps completely along the "height" of the new rail, you're just about right.
  • No, a woodworker can never have enough clamps.
• Wait for your glue to dry. I'd give it a full 24 hours under clamp (unless your glue of choice calls for a longer drying time or the weather is cool).
• Scrape the squeezed out glue from the joints
• Scrape, sand or strip the paint off the rest of the door
• Sand the whole thing smooth
• Paint the whole door with a couple of coats of quality, exterior paint.
• Rehang the door

TL;DR:

You know what... Unless you've already got the tools and woodworking experience, just replace the door with a new one. You can pick up a pre-hung, plain steel door at your local home center for around $100. You can get a "fancy" one with a window and and internal sliding blind for under $200. If you go to a nice wood door, it'll probaly be $400 and up. You'll spend more than $200 on a cope and stick bit set and a router capable of spinning them, plus all the other tools that you probably don't own yet.

Of course...

If you want to get into woodworking, this is the perfect excuse to buy the tools. Just know that this isn't a perfect beginner's project.