The quote by Sheldon from Big Bang Theory is misguided, as the limbic system is not part of the "reptilian brain", it is part of the next development, the paleomammalian complex.
The Reptilian Complex
The R-complex consists of the brain stem and the cerebellum. Its purpose is closely related to actual physical survival and
maintenance of the body. The cerebellum orchestrates movement.
Digestion, reproduction, circulation, breathing, and the execution of
the "fight or flight" response in stress are all housed in the brain
stem. Because the reptilian brain is primarily concerned with physical
survival, the behaviors it governs have much in common with the
survival behaviors of animals. It plays a crucial role in establishing
home territory, reproduction and social dominance. The overriding
characteristics of R-complex behaviors are that they are automatic,
have a ritualistic quality, and are highly resistant to change.
The limbic system, the second brain to evolve, houses the primary centers of emotion. It includes the amygdala, which is
important in the association of events with emotion, and the
hippocampus, which is active in converting information into long term
memory and in memory recall. Repeated use of specialized nerve
networks in the hippocampus enhances memory storage, so this structure
is involved in learning from both commonplace experiences and
deliberate study. However, it is not necessary to retain every bit of
information one learns. Some neuroscientists believe that the
hippocampus helps select which memories are stored, perhaps by
attaching an "emotion marker" to some events so that they are likely
to be recalled. The amygdala comes into play in situations that arouse
feelings such as fear, pity, anger, or outrage. Damage to the amygdala
can abolish an emotion-charged memory. Because the limbic system links
emotions with behavior, it serves to inhibit the R-complex and its
preference for ritualistic, habitual ways of responding.
The limbic system is also involved in primal activities related to food and sex, particularly having to do with our sense of
smell and bonding needs, and activities related to expression and
mediation of emotions and feelings, including emotions linked to
attachment. These protective, loving feelings become increasingly
complex as the limbic system and the neocortex link up.
Also called the cerebral cortex, the neocortex constitutes five-sixths of the human brain. It is the outer portion of our brain,
and is approximately the size of a newspaper page crumpled together.
The neocortex makes language, including speech and writing possible.
It renders logical and formal operational thinking possible and allows
us to see ahead and plan for the future. The neocortex also contains
two specialized regions, one dedicated to voluntary movement and one
to processing sensory information.
The Triune Brain Theory is a cross-species evolutionary study with the developmental processes underlying them (I borrowed a line from (1) to use as a definition, as I thought it summed it up well). As mentioned in other answers, it is an oversimplification and doesn't match work across all species.
As opposed to the study of Embryology.
(1) Mammalian Brain Evolution
Morphological Evo-Devo Group
Neural System Development