I am a TERF but I do want to listen to trans people and too many of them describe this 'feeling' to dismiss it as nothing so the above is attempt to try and make sense of that feeling and look at the research we do have - it cannot just come from nowhere IMO - but whether it is biological or cultural, we just don't know - but more likely an interaction of both
It's a mistake to assume that gender dysphoria is a single, definable condition. Actually there are a whole lot of different mental health conditions being subsumed under the banner of 'trans'. For instance, men and women transition for very different reasons, and even among men, gender dysphoria has different aetiologies.
The main ones for men are Blanchard's homosexual/non homosexual typology, whereby homosexual trans males are those who felt feminine/different from other boys from a very young age, and heterosexual/bisexual trans-identified men are actualising a sexual fetish called autogynophilia, where they get aroused by cross-dressing and thinking of themselves a 'woman'. This typology is very well documented clinically (and on the countless 'sissy' porn sites which cater to autogynophiles
).
The recent upsurge in teenage girls and young women identifying as trans (whereas female transitioners were very rare up until about a decade ago) seems to fit within a pattern of other body dysmorphic conditions that predominately affect young females, like anorexia. This latest manifestation appears to be driven by a reaction to cultural lesbophobia and misogyny, as well as drawing in lots of young women who have Autism Spectrum Disorders. There's a strong social contagion aspect to it too, with multiple girls within one peer group sometimes suddenly deciding they are 'trans'.
The mystical idea that people can have feminine brains/souls in male bodies and vice versa is appealingly simple and goes right back to the 19th century idea of 'inverts' (which was the medical 'explanation' then for homosexuality), but the reality is a lot more complex.
This article by two sexologists gives a good overview. I have some disagreements with the way they ignore the political and cultural context of 'gender' and treat it purely as a medical issue, but it does provide a useful summary of what the clinical data reveal:
Gender Dysphoria is Not One Thing