And why can’t they become Jews?
My understanding is that the term mamzer, referring to the offspring of either an adulterous or an incestuous relationship, is applicable only to Jews (male or female) but not to non-Jews. Thus, non-Jews who are offspring of these relationships can marry Jews provided they convert to Judaism, according to Orthodox Jewish law. The Jewish mamzer, however, and their descendants, are forbidden to marry a Jew based on the proscription found in Deuteronomy 23. No other legal or social restrictions apply to them as Jews. Nonetheless, the Talmud (Mishnah) offers some “solutions” to the personal status of mamzerim and their descendants, and, not surprisingly, it is a complicated legal matter. In the Messianic era, all will be rectified; but meanwhile Reform Judaism simply ignores the whole issue by not accepting the restrictions to begin with. For more insight, I would suggest you peruse the websites Jewish Virtual Library and My Jewish Learning.
But why? I know that, but why can’t they do things like marry other Jews, just because they are ‘bastards’?
My understanding is they are Jews. They just have issues with illegitimacy or arising from incestuous relationships. Their treatment in modern Jewry is dependent upon their individual sect and status.
They are not, technically speaking, bastards since, as I mentioned, they have complete legal rights except for marriage. The “why” regarding the latter restriction depends on one’s interpretation of the Deuteronomy verse I referred to. A literal interpretation would seem to exclude them from entering into the Congregation of G-d (at least during pre-Messianic times) by means of marrying another Jew. However, this interpretation is contrary to the Jewish law that only the guilty party be punished for their sins, not innocent bystanders, for example, children. Enter the Talmud, the (codified) Oral law, which is designed to interpret more profoundly the Torah or Written Law by filling in the blanks. And therefore there is discussion of this issue, including possible remedies such as not requiring an investigation of the status of the individual who is a mamzer nor their children. But, I reiterate, Jewish law is complicated as is secular law.