I think you'll struggle to get a glossy stained finish on the EX1. The body pieces have their grain patterns at very different angles to each other. They are also joined at an angle (compared to the centre line of the neck), rather than parallel to it (as in your example photo). You can fill the small gaps at the neck join with filler, but it will be hard to make the colour match and not be noticeable.

As long as the body pieces are securely glued together, then any gap between pieces can again be filled in and sanded flat.

My EX1 was similar with regards to the angled body blocks and the neck gaps, so I went for a solid colour finish. This allowed me to get a very smooth neck/heel joint indeed.

It's probably not what you want to hear, but it's the best way to get a good looking guitar from the kit. The EX1 kit really isn't put together at the factory properly to get a really good looking natural finish that's anything like an original korina-bodied Explorer. A pickguard will help, but the standard ones will have a hole in for the pickup selector switch, and the kit has the selector switch positioned between the volume and tone knobs.

You can, (like I and several others have done), rout out your own selector switch channel and fill in and redrill the centre hole to be another volume knob position, but it's more work and another reason not to use a see-through finish.

You can certainly have a go at staining to try and achieve a similar finish to your example. You'll first need to grain fill the mahogany, sand it flat and then use a pale yellow-brown stain on top of that. As the body is mahogany rather than korina, you'll find it hard to match the colour in the photo, as the mahogany is naturally more red-brown than yellow-amber. You can stain a wood darker, but it's very hard to make it lighter. Possibly try a white stain first.

Then you'll need a good clear coat on the top, possibly one or two layers with an amber tint in, and then fully clear lacquer over that. Using amber tint would depend on how light the body was after staining, as you won't want to make it too dark. Gibson would use a nitro lacquer, but acrylic or polyurethane would be cheaper and easier to get hold of.