What should I do to improve this snippet
I noticed that when moving through the OpenFileDialog to find files it
  felt sluggish. Is that because I had to set [STAThreadAttribute] just
  to get it to open?


That is a StackOverflow or GameDev.StackExchange kind of question.

Refactoring the code like this could be a start:

private void loadMapToolStripMenuItem_Click(object sender, EventArgs e)
{
    var openFile = new OpenFileDialog
                       {
                           Title = "Select a File",
                           InitialDirectory = Application.StartupPath + @"\Content\Map\",
                           Filter = "MAP Files (.MAP)|*.MAP",
                           FilterIndex = 1,
                           Multiselect = false
                       };

    if (openFile.ShowDialog() != DialogResult.Cancel)
    {
        try
        {
            string otherMapName;

            LoadMap(TileMapper, openFile.FileName, out otherMapName);

            txtMapName.Text = otherMapName;
        }
        catch
        {
            //TODO: display an error message box?
            System.Diagnostics.Debug.Print("Unable to load map file");
        }
    }
}

//TODO: why not move this method inside TileMapper?
private void LoadMap(TileMapper tileMapper, string fileName, out string mapName)
{
    mapName = null;

    // Example file - _W25_H25_SET_GL_test.Map
    tileMapper.LoadMap(new FileStream(fileName, FileMode.Open));

    mapName = Path.GetFileNameWithoutExtension(fileName);

    //get info stored in file name
    string[] nameParts = mapName.Split('_');
    string[] widthPart = nameParts[1].Split('W');
    string[] heightPart = nameParts[2].Split('H');

    tileMapper.MapWidth = int.Parse(widthPart[1]);
    tileMapper.MapHeight = int.Parse(heightPart[1]);
}