I'll start by saying that I can't prove it to you. For all you know, your experience constitutes the entirety of existence. That's not inconceivable.
All you can really do is use inductive reasoning. For one thing, you know that you have reactions to external stimuli, and those reactions are driven by your experience. For example, when you are hit, you experience pain. And when you experience pain, you express it in certain ways. Others follow a similar pattern. External stimuli that would cause you pain, when imposed on others, cause them to express themselves in a similar manner as you would. The only thing missing from your perspective is their pain. However, you know that, for yourself, that pain is a necessary link in the causal chain from the external stimuli to the pained reaction. So you might infer that pain is part of the process for them too, even though you can't observe it.
In science we often deduce things that are hidden from our senses when they're the best explanation for something. For example, the concentrated positive charge and dispersed negative charge of an atom was the best explanation for many experiment results, such as the Rutherford scattering experiment. So we deduced the structure of an atom without being able to see it directly. We simply accept it as the best explanation for the structure, until a better explanation comes along.
Others' reactions to painful stimuli, and its consistency with your experience is one of many methods in which you can infer that something you don't observe - the others' pain - is present. There are other examples. You know, for one thing, that if a concept you're unfamiliar with is introduced to you and after a short time you make an inference that couldn't be made without understanding of said concept, then you necessarily had the conscious experience of deliberating on this concept. So if you introduce a concept to another individual, and after a short time, they make an inference that requires understanding of said concept, you might infer that the necessary step of conscious deliberation was involved. It's not necessarily the case, but it is another hint that others have a conscious experience.
If things like this doesn't convince you that there are other conscious experiences besides your own, there are still other ways that there is still existence beyond your own experience. For example, as far as you're aware, humans can only come into existence through the birthing process performed by other humans, before that human is born. You are human, and you know that you didn't have conscious experience before you were born. Inductive reasoning would lead you to conclude that you likely came into existence by a process performed before you were born, which implies that events happened independent of your conscious experience. However, this is still inductive reasoning; it's possible that you're the sole exception to this rule, despite what your parents tell you. Similarly, you observe that existence carries on when others die, so you might infer that the same would be the case for yourself. Again, though, you might be the sole exception.
There's plenty of other ways you can infer that existence occurs independently of your experience, but only through inductive reasoning. Again, existence outside your experience is a phenomenon you don't observe, and, as with scientific phenomena, we can only infer things about the world that we don't observe through inductive reasoning. But we cannot disprove solipsism, so you have to live knowing that it is a possibility, however remote.