First, the "size" of your image is absolutely NOT measured in kilobytes. That is only the file size, which largely depends on the degree of compression you choose. Due to the JPG Quality factor selected, larger JPG files are better quality, and smaller JPG files are lower quality (lower quality is not normally the best choice), but the image size in pixels remains unchanged then.
This first part only addresses that confusion, but the answer is in the next part below.
Any JPG file is standard 24 bit color, which means the size of the image data is 3 bytes per pixel. But the image size is the number of pixels, for example, lets say image size is 4000 x 3000 pixels, which is 12 megapixels, so the uncompressed data size (data size when opened into computer memory) is 12 x 3 = 36 million bytes. However data (in the JPG file) is always compressed in some degree (controlled by the JPG Quality option), so the actual compressed file size might typically be maybe from 1/4 that to 1/12 that size. But this example image size remains the 4000x3000 pixels, regardless of compressed file size.
The 300 dpi is about printing it on paper, and 300 dpi means 300 pixels per inch of print paper. Therefore, to set print resolution (of this example 4000x3000 pixels image size) to be 300 dpi means it will print 4000/300 x 3000/300 = 13.3 x 10 inches in size on paper. This specific print size is simply because the image size is 4000x3000 pixels, and that's how far 300 pixels per inch will stretch when printed on paper.
uncompressed data size: 36 million bytes
JPG compressed data file size: Perhaps 9 to 3 million bytes.
print size at 300 dpi: 13.3 x 12 inches
image size: 4000 x 3000 pixels.
Each of these concerns have different units.
Whoever asked you for 300 dp should have specified the printed size in inches that they seek. Otherwise, 300 has no meaning or use for them. Dpi has nothing to do with image size, which is in pixels. Dpi does affect print size, but which needs to know the image size in pixels to have any meaning.
The asker probably implies they want a large image, probably the original camera image size in pixels. Meaning, they should have requested image size in pixels, or at least the printed size in inches if at 300 dpi.
The terms dpi and ppi used in printing are interchangeable with the same meaning, pixels per inch. There is more that can be said about that, but it is largely not important today, because printer drivers have mostly given up the notion of asking about print quality in "ink drops per inch", which is something entirely different. To bypass that, print quality today is the Good, Better, Best categories.
Using a photo editor, you can simply change the dpi number to be 300 dpi. This is called "scaling the printed size" in inches, which is what dpi does. That will then specify some specific printed size in inches, which may or may not be acceptable, depending on your image size in pixels, and on your purpose for using the image.
Yes, if you change dpi to be 300 dpi and then save the file, yes, it will then say 300 dpi in the file. Some editors today may call it 300 ppi, same thing.
This scaling to 300 dpi does NOT affect the number of bytes, UNLESS you also change either the resampled size in pixels, or change the JPG Quality factor.
It does change the scaled print size inches, to be the number of inches that 300 pixels per inch will print from the pixels that you have.