It appears that Clinton and the Republican majority that actually passed this law both shared a vision of time-limited benefits. It's not terribly well articulated why in the sources I found, but it does seem (as Shadow1024 suggested in a comment) that eliminating a "welfare trap" was the underlying goal.
Quoting from the Wikipedia article you linked:
Conservatives emphasized work requirements and time limits, paying little attention to whether or not families' incomes increased. More specifically, conservatives wanted to impose a five-year lifetime limit on welfare benefits and provide block grants for states to fund programs for poor families. [...]
Both President Clinton and Congressional Republicans emphasized the need to transform the cash assistance system into a work-focused, time-limited program.
It's citing a Northwestern University IPR newsletter for the first para and a (much longer) CLASP paper for the 2nd; the first source cited has slightly more detail on the immediate debates around TANF:
Ron Haskins, who was a senior Republican staff member on
the House Ways and Means Committee at the time, started
by underscoring the bitterness of the debate.
He read a passage from his book Work Over Welfare describing how Congressman John Lewis (D-Ga.) equated the Republicans with
Nazis for their “mean-spirited” bill that “took food out of the
mouths of children.”
“Very few debates in the House have this intensity,” he said,
pointing to a shortage of facts and an abundance of emotion.
Looking back, Haskins noted that “the radical Republican
agenda” consisted of wanting to increase work and marriage
to have the greatest impact on poverty. “And this was what
welfare reform did,” he said.
Haskins listed the five fundamental changes to welfare: ending cash entitlements, creating block grants, instituting five year time limits, implementing strong work requirements, and
“It is very rare that any program in Washington changes to
this degree,” Haskins added.
Some of these changes, such as the five-year
time limit, were highly controversial despite
the bill’s bipartisan support.
[...; and for the view from the opposite side of the isle]
In crafting the Democrat’s legislative proposal, Ellwood said
he and his colleagues defined three core elements: making
work pay, enforcing child support payments as a way of making both parents responsible for raising their children, and
making welfare “a hand up and not a handout.”
While Republicans and Democrats generally agreed upon
and passed the make-work-pay and child support aspects, he
said, they differed greatly on how to transition families to—
and support them in—the world of work. Clinton originally
wanted to provide job training coupled with time limits and
subsidized jobs for those having difficulty finding work. Republicans wanted a five-year lifetime time limit on individuals’
welfare benefits and block grants for the states. In the end,
the Republican vision passed.
The 2nd source says:
President and Congressional Republicans emphasized the need to transform the cash assistance
system into a work-focused, time-limited program. [...]
And give a bit more historical context:
In 1992, presidential candidate Bill Clinton pledged to “end welfare as we know it” by requiring
families receiving welfare to work after two years. In 1994, the Clinton Administration introduced a
welfare reform proposal involving expanded services and requirements intended to increase workforce
participation for single parents, along with a requirement that parents participate in a work program as a
condition of receiving further assistance after a family had received assistance for two years. The
proposal did not advance, and after Republicans attained a Congressional majority in November 1994,
the focus shifted toward the Republican proposal to end entitlements to assistance, repeal AFDC and
instead provide states with block grants.
The 1996 law repealed AFDC and enacted the structure of TANF block grants. The TANF
structure is best understood as a hybrid. States receive a lump sum of money that can be used for an
array of purposes. One purpose is to operate a program of assistance for needy families. A set of
requirements -- e.g., time limits, work requirements, child support cooperation -- apply to families
receiving TANF “assistance,” but not to those receiving other benefits and services funded under the
block grant. [...]
Generally, under TANF, most states developed time-limited assistance programs with a strong
emphasis on work-related requirements. Most states developed programs in which most or all parents,
including parents of very young children, were required to participate in work-related activities; adopted
policies under which all cash assistance could be terminated for a violation of program rules; developed
policies under which family assistance can be reduced or terminated after a time limit of 60 months or
The CLASP paper also attempts to estimate the effects of TANF, but I'm omitting that here since it wasn't the topic of the question.