Help with libcurl .Net

No need to declare FileStream fileStream outside the method using this object. Probably, you never should do it. This way, you're using a local variable instead of the class field. Always prefer local whenever possible. Do in under using block -- it will call Dispose automatically at the end of block even if there was an exception.

using (fileStream stream = new FileStream("aaaaa.txt", FileMode.Create) {
    stream.Write//...
} //auto-call stream.Dispose();

You called Dispose anyway, which is very bad, you could have lost file records because nothing did the flush.

Also, you could better use StreamWriter instead:

using (StreamWriter writer = new StreamWriter("aaaaa.txt") {
    writer.Write//...
    //you can access underlying stream from writer if needed
} //auto-call fileStream.Dispose();

Also, keyword "private" is C# default (for class or structure members -- thanks to Manfred for correction, see his comment below), can be omitted.

[EDIT] This is how to apply the same principle to Easy. First, implement System.IDisposable:

public delegate System.Int32
    WriteData(
       FileStream stream, //added parameter
       byte[] buf, Int32 size, Int32 nmemb, bbject extraData);

class Easy : System.IDisposable {
    public Easy(FileStream stream, WriteData onWriteData) {
        this.Stream = stream;
        this.OnWriteData = onWriteData;
    } //Easy

    public void Perform() {
        //if (this.OnWriteData != null)
        //    OnWriteData(this.Stream, //...
    } //Perform
    void System.IDisposable.Dispose() {
        try { //important to make sure...
            CleanUp();
        } finally { //make sure Stream is always disposed:
            if (Stream != null)
                Stream.Dispose();
        } //exception
    } //System.IDisposable.Dispose
    void CleanUp() { /* ... */ }
    FileStream Stream;
} //System.IDisposable

This is how to use it:

partial class MyForm {

    void MyButtonClickHandler() {
        using (Easy easy =
           new Easy(
                new FileStream("fname.aaa",
                FileMode.Create), OnWriteData)) {
            //..
            Easy.Perform();
            //..
        } //using -- clean-up here
    } //MyButtonClickHandler

    System.Int32 OnWriteData(
        FileStream stream,
            byte[] buf, Int32 size, Int32 nmemb, bbject extraData)
        {
        stream.Write(buf, 0, buf.Length);
        return size * nmemb;
    } //OnWriteData

} //class MyForm

Finally, this is how to use MyButtonClickHandler -- better way:

MyButton.Click += (sender, eventArgs) => { MyButtonClickHandler(); };

I don't know your C# version, there is no lambda for v.2.0, but you still better of with anonymous:

MyButton.Click += delegate( object sender, System.EventArgs eventArgs) {
   MyButtonClickHandler();
};