Lacking specifics, here is a general example of how to populate the "presented lessons" DataGridView and adding/removing a row in the "selected lessons" DGV when selected. At least now you will have a starting point to experiment.

It's usually easier to work with DGV if you make a class to represent a row of data.

class Lesson 
{
    public bool Selected { get; set; }
    public string Description { get; set; }
}

Next you declare lists that contain instances of your Lesson class.

BindingList<Lesson> All { get; } = new BindingList<Lesson>();
BindingList<Lesson> Selected { get; } = new BindingList<Lesson>();

The binding lists are assigned to the respective DGVs when the main form loads. This method will also attach an event for when cell content (e.g. a checkbox) is clicked.

public partial class MainForm : Form
{
    public MainForm() => InitializeComponent();
    protected override void OnLoad(EventArgs e)
    {
        base.OnLoad(e);
        dataGridViewAll.AllowUserToAddRows= false;
        dataGridViewAll.RowHeadersVisible= false;
        dataGridViewAll.DataSource = All;
        dataGridViewSelected.DataSource = Selected;
        dataGridViewSelected.AllowUserToAddRows= false;
        dataGridViewSelected.RowHeadersVisible = false;

        #region A U T O G E N E R A T E    C O L U M N S
        All.Add(new Lesson());
        Selected.Add(new Lesson());
        #endregion A U T O G E N E R A T E    C O L U M N S

        #region F O R M A T    C O L U M N S
        dataGridViewAll.Columns[nameof(Lesson.Selected)].Width = 40;
        dataGridViewAll.Columns[nameof(Lesson.Selected)].HeaderText = string.Empty;
        dataGridViewAll.Columns[nameof(Lesson.Description)].AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;
        All.Clear();

        dataGridViewSelected.Columns[nameof(Lesson.Selected)].Visible = false;
        dataGridViewSelected.Columns[nameof(Lesson.Selected)].HeaderText = string.Empty;
        dataGridViewSelected.Columns[nameof(Lesson.Description)].AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;
        Selected.Clear();
        #endregion F O R M A T    C O L U M N S

        dataGridViewAll.CellContentClick += onCellContentClick;
        dataGridViewSelected.SelectionChanged += (sender, e) =>dataGridViewSelected.ClearSelection();
        addTestData();
    }
    .
    .
    .
}

How to populate the first DGV:

private void addTestData()
{
    // Add a few classes for testing purposes
    All.Add(new Lesson
    {
        Description = "Learn C# with Windows Forms"
    });
    All.Add(new Lesson
    {
        Description = "C# Database Driven WinForm Apps"
    });
    All.Add(new Lesson
    {
        Description = "Learn C# With SQL Server"
    });
    dataGridViewAll.ClearSelection();
}

Now all you have to do is handle the content click and either add the lesson to (or remove the item from) the second bound collection for the lesson that has changed.

private void onCellContentClick(object sender, DataGridViewCellEventArgs e)
{
    switch (dataGridViewAll.Columns[e.ColumnIndex].Name)
    {
        case nameof(Lesson.Selected):
            if(e.RowIndex != -1) // Don't execute is this is the header cell
            {
                dataGridViewAll.EndEdit();
                if (All[e.RowIndex].Selected)
                {
                    Selected.Add(All[e.RowIndex]);
                }
                else
                {
                    Selected.Remove(All[e.RowIndex]);
                }
                dataGridViewAll.ClearSelection();
            }
            break;
    }
}