I'm not very strong in group theory, so could someone please explain
in simple terms what it means for the left handed parts to transform
as triplets and right handed parts to transform as singlets? How would
you go about writing down such terms in a Lagrangian?
Groups are abstract. They have elements that can be "multiplied" and they have other properties (for example, every element has an inverse, etc). Representations of groups are concrete. For example a 3x3 matrix could be used to represent an abstract group element. Representations of groups can be multiplied, for example matrices can be multiplied by matrix multiplications.
Say, for example, that you have three abstract elements of a group called "a", "b", and "c". And say that
Say, also, that you have a specific representation of those three above-mentioned group elements, the representation being 3x3 matrices called: R(a), R(b), and R(c). Then, you want the same multiplication rules to hold for the representation as for the group itself, namely:
where the $\cdot$ now means matrix multiplication.
For every element in the group there should be some element in the representation of the group, and all the multiplication rules that hold for the elements of the group should hold for the elements of the representation.
The representation of the group does not have to be 3x3 matrices. There are an infinite number of different representations of a group.
One interesting representation is 1. I.e., the representation in which every element of the group is represented by 1. This is trivial, right? Yeah, that's why it is called the "trivial representation". But it works because, suppose that $a \to 1$, $b \to 1$, and $c \to 1$. Then
a\cdot b = c \to 1\cdot 1 = 1
which is true.
So every group has a trivial representation.
When we say something "transforms under" some representation of the group, we mean that it gets multiplied by some representation of the group under the group transformation. If something transforms as a singlet it is transforming under the trivial representation. I.e., it gets multiplied by 1. I.e., it doesn't transform.
On the other hand, objects transforming under the fundamental representation of SU(3) get multiplied by 3x3 special unitary matrices. I.e., objects transforming under this representation are like 3x1 column vectors. I.e., triplets. There seems to be a commentator on your question that is confusing this issue.
There are other representations beside the trivial and the fundamental. As I said earlier there are an infinite number of representations. In the case of SU(3) there are also an infinite number of irreducible representations... but a discussion of this point is beyond the scope of this answer.