『えをかく』means the infinitive/dictionary form for "draw pictures."

And your confusion is probably coming from the fact that in English, the "infinitive forms of verbs" in vocabulary lists in phrase books,  language textbooks, etc almost always have "to" in front of them to mark them as the infinitive form.

I'm assuming that you must have studied French, right?

So, let's take the word "study" in English and French as an example. The English word "study" can be a nous as in "study of English literature" and a verb as in "I study Japanese everyday."

But if you simply look at the word "study" alone in these two sentences, you cannot tell whether they are a noun or a verb because the forms are exactly the same "study."

While on the other hand, in French, they will be something like;

"étude de la littérature anglaise" and "J'étudie Japonais tous les jours" 

※Sorry, my French is rusty, so I might be wrong.

So the forms are different "étude" and "étudie". And on top of that, the dictionary form for the verb "study" is "étudier." So, the forms are all different in French, but they are all translated as "study" in English. So in order to distinguish "study" as a noun and "study" as a verb, people often put "to" in front of the word "study" to indicate that the word "to study" is a verb.

Something similar is happening in the case of your Japanese sentence.

The phrase 『えをかく』is an infinitive/dictionary form of "draw pictures". But if you want to say something like;

"I like TO draw pictures." you need to say 『わたし は えいをかく "の" が すきです。』to say "TO draw pictures" or "drawING pictures". You don't say "I like draw pictures" in English, right?

So, grammatically speaking, "えをかく+の"  or "TO draw pictures" in this sentence is not a noun nor an infinitive. It is called "noun phrase"; a phrase that functions as a noun in a sentence.

え　is a noun. え means a picture.

かく is a verb. We use かく both mean write and draw.

描く mean draw.

書く mean write.

I wish this explain help you to understanding Japanese.

And then, another confusion is probably coming from the fact that there is a thing called "homophones.""

"Homophones" are two or more words that have the same pronunciation, but have different meanings such as "air" and "heir," "hair" and "hare", "flower" and "flour" etc.

And the word for "to draw" and "to write" in Japanese are homophones;

書く(kaku)--- to write

描く(kaku)---to draw 

To distinguish these two homophones, in some textbooks, they might say something like;

"描く(kaku)" ---to draw pictures. 

However, the word "描く(kaku)" per se does not contain the part "pictures". So if you want to say "draw PICTURES", you still need the word "えを"

The phrase "おえかき" means "picture-drawing activity." So a kindergarten teacher might say to his or her kids

"つぎは おえかき の じかん ですよ！" which means "Now it's time for drawing activities!"

So you want to express the idea of 

"I used to draw pictures of birds and squirrels when I was a child"

in Japanese, right?

OOOh, I was sure 描 means cat. ! note to myself but maybe it's useful for anyone, cat is 猫