The two cases mentioned in the linked podcast are those of Crystal Mason and Rosa Maria Ortega, who were both convicted of voter fraud under different circumstances.
I'm going to firstly focus on the case of Crystal Mason, who was convicted after voting in the 2016 presidential election whilst ineligible to vote.
She was able to be identified and charged because as she did not appear on the list of registered voters, when she went to vote at her polling place, she filled in a provisional ballot paper. These were introduced by the Help America Vote Act in 2002, and are used to ensure that voters who think they should be able to vote, but do not appear on the list for administrative reasons, can still cast their vote on the day. You can read more about Texas's specific implementation here.
Provisional voting is designed to allow a voter whose name does not
appear on the list of registered voters due to an administrative error
to vote. The provisional voting process involves an affidavit that (1)
the voter must complete stating the reasons he or she is qualified to
vote; and (2) is used if the voter’s registration cannot be verified
by the polling place election officials OR if a voter (a) does not
possess one of the acceptable forms of photo identification listed
above, and a voter can reasonably obtain one of these forms of
identification or (b) possesses, but did not bring to the polling
place, one of the seven forms of acceptable photo identification
listed above, or (c) does not possess one of the seven forms of
acceptable photo identification, could otherwise not reasonably obtain
one, but did not bring a supporting form of identification to the
The provisional voting process requires the voter to visit the voter
registrar’s office within six (6) calendar days of the date of the
election to either present one of the above seven (7) acceptable forms
of photo ID OR if the voter does not possess, and cannot reasonably
obtain an acceptable form of photo identification, execute a
Reasonable Impediment Declaration and present one of the acceptable
forms of supporting ID, OR, if applicable, submit one of the temporary
affidavits (e.g., religious objection or natural disaster) OR, if
applicable, qualify for a permanent disability exemption, in order for
the provisional ballot to count.
The voter-marked provisional ballots are kept separately from the
regular ballots, and the voter’s records will be reviewed by the
provisional voting ballot board (the early voting ballot board), to
determine if the ballot is to be counted or rejected. If applicable,
the voter registrar will conduct whatever research is necessary to
determine whether the voter is or should have been registered in the
precinct in which the voter cast the provisional ballot and will pass
this information on to the ballot board to assist it in making the
decision of whether the provisional ballot must be counted.
Provisional voters will receive a notice in the mail by the 10th day
after the local canvass advising them if their provisional ballots
were counted, and if they were not counted, the reason why.
Mason was ineligible to vote due to previously having been convicted of a felony; tax fraud, and being on supervised release. Texas prohibits felons from voting until they finish their sentence entirely. As a result, she was charged and convicted of voter fraud, as the provisional ballot with attached affidavit served as evidence that she had attempted to vote. The Guardian reports:
She did not read the small print of the form that said that anyone who
has been convicted of a felony – as she had, having previously been
convicted of tax fraud – was prohibited from voting under Texas law.
Her sentence was upheld by a Texas appeals court in March 2020.
Secondly, the case of Rosa Maria Ortega. Ortega is a Mexican national who registered to vote despite being ineligible, and later confessing to officers while being secretly recorded to having voted previously. According to USA Today:
Birdsall characterized Ortega as a poorly educated woman who, as a
lawful permanent resident all of her adult life, was unaware that she
was not permitted to vote. Her indictment in November 2015 followed a
series of actions she revealed to elections officials and law
After moving from Dallas to neighboring Tarrant County in late 2014,
she attempted to register to vote but indicated on her application
that she was not an American citizen. When her application was
rejected, she called election administrators and was told that the
reason for the rejection was that she had checked the "no" box for
citizenship. Ortega explained that she had been able to vote in Dallas
County and resubmitted her voter registration, this time indicating
she was a citizen.
Several months later, Ortega was visited on her front porch by two
investigators from Paxton's office. They secretly recorded Ortega as
she said she checked the box indicating she was a citizen because she
had previously encountered no trouble voting in Dallas County.
It was Ortega's poor luck that she had just confessed to illegal
voting in a state where elected officials made examples of those they
deemed contributors to voter fraud.
So in the two cases mentioned, there were specific circumstances which provided the court with enough evidence to convict the individuals of voter fraud.