The somewhat incoherent argument aside .... she is actually trying to tie brexit, already a major change, to a second major change but this time internal? Not sure how the EU would oppose a socialist setup anyway given there are socialist member states just to the north of her.
"Socialist" means something different in the USA, I think, from what it means in the rest of the world (certainly from what it means in Europe, anyway). Most of us on this side of the Atlantic would regard the Scandinavian countries as models of social democratic mixed economies rather than socialist.
EU rules on competition and non-tariff trade barriers prevent, or at least constrain, certain types of state aid and subsidies to businesses. So national governments can't give preference to domestic suppliers and contractors in procurement and neither can they readily (if at all -- I just don't know) give state aid to failing industries to prevent unemployment, and neither can they readily nationalise whole sectors of industry and give them preferential treatment compared to their EU competitors.
Nationalisation -- public control of industry -- used to be a key policy of the Labour Party. Clause 4 of the old Labour Party constitution used to list as an objective of the party
To secure for the workers by hand or by brain the full fruits of their industry and the most equitable distribution thereof that may be possible upon the basis of the common ownership of the means of production, distribution and exchange, and the best obtainable system of popular administration and control of each industry or service
This was finally dropped in the 1990s by Tony Blair, but many Labour activists -- including, I think, Jeremy Corbyn, who doesn't seem to have changed his views much since the 1970s -- would like to return to those policies. EU membership would certainly make that difficult under current rules.
That, I think, is what she means by "Socialist."