If the set of odd numbers is a subset of $\mathbb N$ then surely it is smaller than $\mathbb N$

It depends on your definition of "larger". Once you make that precise, you'll answer your own question.

Imagine that you didn't know that $\mathbb O$ is a subset of $\mathbb N$, or that they have any elements in common. Imagine that you couldn't even inspect the properties of the elements of $\mathbb O$ and $\mathbb N$. Imagine that you didn't even recognize those elements as numbers. To paint a picture, imagine that someone takes a copy of $\mathbb O$ and a copy of $\mathbb N$ and encloses all of their elements in whimsically colored hardwood boxes.

Now, all you know is that $\mathbb N=\{a,b,c,d,\ldots\}$ and $\mathbb O=\{A,B,C,D,\ldots\}$, where $a$ denotes the teal-striped rosewood box, etc. Maybe $a$ contains the number $0$. Maybe $B$ contains the number $3$. But you have no way of knowing that.

Under these restrictions, how will you compare the sizes of $\mathbb O$ and $\mathbb N$? Can you come up with any justification that one is larger than the other?

For the following lists,

$$1,2,3,...$$

$$1,3,5,...$$

erase the labels, and replace them by dots:

$$\small{\bullet,\;\bullet,\;\bullet,\;...}$$

$$\small{\bullet,\;\bullet,\;\bullet,\;...}$$

Now which list has more elements?

The key idea is that for comparing the "sizes" of two sets, labels shouldn't matter. What matters for determining equal size is one-to-one correspondence.