How do I remove the background from an image?

In GIMP you can convert the image to black and white. It's a feature called Desaturate.

Now that you have it in B&W then increase Brightness and Contrast in a way that the background becomes bright white and the letters are deep dark.

just like these example:

In that state will be easy to select all the white color with this tool

Be sure that the threshold of selection is near to zero to select only a single color and select feather edges.

Add a transparent layer (Layers > Transparency > Add Alpha Channel)

Clear (Edit > Clear or Del) the selected color and you will have a transparent background.

If you want to have a new background, add a Layer and put the color in the new layer.

I would advise vectorising the logo with Inkscape. Open Inkscape, select File > Import... to import the logo, select Path > Trace Bitmap... to trace the logo to a vector, adjust the settings to create a clean vector (tick the Live Preview checkbox to see how your final vector will look) and click OK when you're happy with the settings. The vectorised logo will be placed over the bitmap logo, so move it out of the way and delete the original bitmap from the Inkscape file. Save the Inkscape file (the format is .svg, for a vector graphic file).

As this is a vector graphic, you can now easily change the colours, style of the lines, and shape of the logo, and can export it at different sizes and resolutions without it appearing blurry or pixelated.

With Paint.NET use Brightness/Contrast like this:

If you need it transparent, use the wand with 0% tolerance and flood mode global, click on a white region and then press Del to delete the white background.

I saved your image and opened in Gimp. I used the Select by Color tool with a threshold of about 125. I selected the dark green in the center letters. I copied it and pasted into a new image. Saved as xcf and exported as png. There's some pixelation (this is not a vector approach as described by others) and the shading around the medium sized text that looks like (forgive me for this) Chinese characters is not clean. Otherwise I think it's ok.

Here's the png:

Edit: Just realized that the "Shandong" means that they probably ARE Chinese characters.

You can do this using Paint by saving it as a low-quality Bitmap-image. You should click on File>Save As and then select Monochrome Bitmap, 16 Color Bitmap or 256 Color Bitmap as file types. Doing this makes there be fewer colors in the image that can be used and therefore changes the colors to the closest of these colors. Which format you should choose depends on the image, in your case using 16-color bitmap would give this:

I admit, like that it doesn't look so nice, but there are several easy ways to fix it so that it looks nicer:

Since there are very few colors, you can easily use the fill tool to change the green to gray for example. You can also use the eraser to remove the gray spots in the background.

Or if you want it to go even faster, you can totally replace a color with another by selecting the color you want to replace as color 2, cutting out the whole image with CTRL+A and CTRL+X, filling it with the color you want to replace with and pasting the image with the "transparent selection" option activated.

In your image, after replacing (192; 192; 192) gray with white and (0; 128; 128) green with (128; 128; 128) gray, it gives this result: