Well I am sure there are many ways but maybe this will help.
EDIT: All below changed to VB.Net for the OP.
First create a new property in your settings file
1) Right click your project in the solution explorer
2) Click on properties
3) Click on settings tab
4) Add a new property called CheckedListState and set its type to string
Then use below code (modify for your use)
Private Sub Form1_Load(sender As Object, e As EventArgs) Handles MyBase.Load
Dim checkedStates As String() = My.MySettings.Default.CheckedListState.Split(New Char() {","c}, StringSplitOptions.RemoveEmptyEntries)
For i As Integer = 0 To checkedStates.Length - 1
CheckedListBox1.SetItemChecked(i, Boolean.Parse(checkedStates(i)))
Next
End Sub
Private Sub Form1_FormClosing(sender As Object, e As FormClosingEventArgs) Handles MyBase.FormClosing
My.MySettings.Default.CheckedListState = ""
For i As Integer = 0 To CheckedListBox1.Items.Count - 1
My.MySettings.Default.CheckedListState += CheckedListBox1.GetItemChecked(i).ToString() & ","
Next
My.MySettings.Default.Save()
End Sub
Ask me if you have any questions about this code and I will try to help you figure it out.