
To use the smart patch tool, follow the following steps:
1: open the photo you want to fix in Krita.
2: select the smart patch tool in the toolbox. Its icon appears to be an X cross made out of two bandages. It's found close to the bucket fill, gradient and eyedropper tools.

3: click and drag to "paint" over the unwanted object in the photo.
Observation: depending on the resolution of your photo and the size of the area you created this can take a long time to finish processing (in terms of minutes long).
4: there is no fourth step because the smart patch tool already fixed your photo. All you have to do left is saving the result.
Tips, Tricks, and Workflows
Now that we already know how to fix photos, some important tips and tricks to be aware about and to use this amazing tool to its full potential.
Using Smaller Sizes
In general, the smart patch tool takes a really long time to smartly patch anything, so you'll probably want to make its radius smaller before painting over the image.
You can change the size of its "brush" by changing the size in the toolbar like in the normal brush tool.
Fixing "Ghost" Patches
If you are trying to patch something particularly large or something that is in proximity of something else, there is a good chance the smart patch tool will patch things wrong.
Normally, what the smart patch tool does is try to figure out the texture it should fill the painted area with that matches the surrounding pixels. The problem is that if there is an object next to the painted area, the tool will think that object is part of the "texture" of the surrounding pixels and will try to copy it or part of it to fill the painted area with it. This creates a "ghost" copy of the other object in the area of the object you are trying to get rid of.
A workaround in this case is to split the large object into smaller parts. Instead of trying to patch a long or tall object all at once, try to patch just the top part first, and if it works, continue until you have patched everything. This workflow is more fluid since it takes less time for the smart patch tool to patch a smaller area and you can tell sooner whether it's working properly or not.
Preventing Spill Over
The smart patch can affect areas of the image beyond the painted area. If you want to prevent this from happening one quick way is to select the area that you don't want the smart patch tool to affect, invert the selection (Select -> Invert Selection on the menubar), and then use the smart patch. Since in this case the protected area isn't selected, it won't be affected by the smart patch operation.
Multi-Step Patching
In some cases, the smart patch tool gets some of its patching right, but not all of it. You don't need to start from scratch in this case.
Simply select the area that it got right, copy it, undo the smart patch, paste what you selected, erase the parts it got wrong, and then merge the pasted layer with your photo.
Now that you have a "half-patched" image, patch the unpatched parts, and repeat until it's all patched.
Blurriness
The smart patch tool is very good at getting some textures right, but in some cases it makes the texture very blurry. This can be particularly noticeable when patching blemishes on skin.
There is a way to decrease the blurriness. In the Tool Options docker, change Patch Radius to 2px and Accuracy to its maximum.
With these settings you can probably fix most skin blemishes using only the smart patch tool. The smart patch tool isn't omnipotent, however, and if you use too much its flatness will obviously substitute the skin's natural texture. In these cases, you will need to use a different too like a clone brush to re-apply the skin texture over flattened areas.
Tip: conversely, if an area is blurry and you're getting "edges," you should increase the Patch Radius instead.

Note: I tried retouching a photo using the smart patch tool to see how well it works in practice. It works great for removing small dots on the skin, but it can't help with larger areas. Although you can easily retouch with it without changing the face's structure, it can't patch the skin texture across large areas, and even the small areas it does patch end up looking a bit "flattened" and "smoothed out." It's faster and more precise than using the clone tool since all you need to do is draw over and it automatically patches the area, but it's certainly not instantaneous specially if you're trying to remove a hundred freckles from a person's face one by one for some reason. Additionally, it can't do much if the area you want to fix is under hair, be it hair of the head or hair of the eyebrows. For very small dots next to details (e.g. folds on eyes), setting the brush size to 1 or 2 pixels and not painting over the area entirely but simply crossing it appears to work. I assume in most cases this tool can be more than enough, but in some scenarios you will still need to use other tools to counter its limitations.

Short Strokes
When work with many small details, it's possible to use the smart patch tool like an actual brush and simply paint over tiny areas quickly and in succession. The only works if the area is smart enough that you computer can process that smart patch before you start the next stroke.