Trans women are women

For any but the most narrow reproductive definitions of "woman", there is no reason to think that "trans women" are not a subgroup of the broader category "women". That's why the phrase is "trans women" and not "trans [something else]".

The Basics

A trans woman is a woman who was assumed at birth to be male, based on superficial physical characteristics. "Trans" is short for "transgender", a term which means "having a gender identity different from that which was assumed at birth". (Its opposite, which applies to the overwhelming majority of humanity, is cisgender or "cis" for short. A "cis woman" is therefore an adult human who was (correctly) assumed to be female at birth.)

• Objection: Women have uteruses and ovaries.
  • Reality: This is usually true, but there are many exceptions... and it's also irrelevant to the discussion.

While such assumptions are correct the overwhelming majority of the time, they are occasionally incorrect to some degree. When such assumptions (and subsequent social role assignments) turn out to be incorrect, the result is a transgender person -- someone who has been perceived and raised as a gender which differs from the one with which they identify.

Here's the basic science:

• Gender is determined very early in life, possibly before birth.
• Gender is immutable.
• Gender is a multidimensional continuum, not a binary.
• Most of the time, most of the elements of gender align into one of two binaries (female or male) – but not always.
• When someone's gender identity is significantly at odds with their assumed gender, we call them transgender.

The word "identify" in that sentence unfortunately opens the door for a lot of misunderstanding about how gender works and why identity is the most important element, superceding any easily-measurable objective qualities.

• Objection: If we allow people to decide their own gender, it will cause serious social problems.
  • Reality: There's actually no evidence for this.

The example most commonly brought up is men declaring themselves to be women so that they can enter women's bathrooms for ogling purposes, but this is a myth. The only known cases of men pointedly entering women's restrooms have been right-wingers trying to make a point. It's far more common for trans women to be abused in men's rooms when they are forced to use them – so allowing trans women to use women's rooms prevents harm without causing any.

Other women's spaces which are sometimes brought up are not generally governed by law, so those are matters to be decided more locally.