If you're on a tight budget and want to get the best "bang for the buck" you need to select the phone that has a camera with strengths in the areas you need them the most while letting go of other features or capabilities that won't affect the kinds of photos you wish to create. Which is more important, sensor size or focal length? Image stabilization or focusing speed? The answer to such questions will vary highly depending on exactly what you want to photograph. Do you want to capture an image of a sculpture in a dimly lit room or do you want to capture your kid in the outfield the instant he makes the home run stealing catch over the outfield wall?
So what is a person such as yourself supposed to do when selecting a phone based on the quality of the camera it includes? You don't know enough about photography to make an informed decision on what is more important for a camera's capability based on the specific types of photos you wish to take. You may not even know what kinds of photos you will eventually become attracted to taking the most. The only way you're ever going to learn how to decide which camera is better for you is to start shooting with any camera and begin to learn a) what kinds of photos do you wish to take and b) what technical features are most important to successfully capture such photos.
So why do the phones with better cameras cost more?
Photography is concerned with capturing light. Although creative photography is concerned with much more than the physics of light and how to record it, the physics of light is still the starting point for every photograph.
While it may be true that technology is rapidly advancing and it may even be true that some of the best smartphones can, under certain conditions, produce photos that are "not far behind their DSLR counterparts," as long as light follows the same laws of physics that it has always followed then cameras with lenses with larger entrance pupils casting light circles onto larger sensors or film will have an inherent advantage over cameras with smaller entrance pupils and sensors.
Any technology that can be applied to a smartphone to allow it to perform better in collecting light can also be applied even more so to those cameras that can collect even more light under the same conditions. In situations where there is more than enough light for either, the difference may be hardly noticeable. In other cases where light is more scarce and collecting every single photon possible can improve the final result then the camera that can collect more light has the potential to allow the photographer to create a better final result.
So the question really isn't, "How much do I have to spend to get a smartphone that takes pictures just as good as a DSLR?" The question is more along the lines of, "How much do I have to spend to get a smartphone that takes pictures that are good enough for me?" The problem with that question is there is no correct answer. The answer will depend both on how good is good enough in terms of the technical capabilities of a camera to produce a certain image as well as what type of photos one wishes to take.
In general, though, the reason the cameras that take better pictures in more demanding situations cost more is because it costs more to create such cameras. Ratings at review sites such as DxO Mark place a premium on how much light a given camera or lens can collect under strictly controlled conditions and how efficiently that camera can transform the light collected to an electrical signal. The camera that can produce the most usable signal from the least amount of light will be rated higher by such an evaluation. The two things that most influence the ability to collect light are the size of the lens' entrance pupil and the size of the sensor recording the light projected on it by the lens. Technological advances can improve exactly what percentage of the light falling on an image sensor is converted to electrical signal. But technology can't improve on how much light falls on a given amount of surface area of a sensor from the same lens collecting the same amount of light and projecting the same image circle.
The price of an imaging sensor rises exponentially in relation to the surface area of the sensor. There are several technical reasons combined with the economics of manufacturing why this is so. The cost to produce a lens that can collect twice as much light as another lens requires roughly four times the volume of material used to produce the lens elements as the other lens. A lens element is three-dimensional and must be increased in all three dimensions in order to double the surface area collecting light and also maintain the same amount of refraction of that light. As lens elements grow larger, though, the slightly different amounts of refraction the various wavelengths of light are bent as they pass through the lens becomes more significant. So lens elements to correct such problems as chromatic aberration must be added to the optical formula, increasing the cost even more.