Ошибка cs0106 модификатор public недопустим для этого элемента

When using explicit implementation of an interface, the members are forced to something more restricted than private in the class itself. And when the access modifier is forced, you may not add one.

Likewise, in the interface itself, all members are public. If you try to add a modifier inside an interface you will get a similar error.

Why are explicit members (very) private? Consider:

interface I1 { void M(); }
interface I2 { void M(); }

class C : I1, I2
{
    void I1.M() { ... }
    void I2.M() { ... }
}

C c = new C();
c.M();         // Error, otherwise: which one?
(c as I1).M(); // Ok, no ambiguity. 

If those methods were public, you would have a name-clash that cannot be resolved by the normal overload rules.

For the same reason you cannot even call M() from inside a class C member. You will have to cast this to a specific interface first to avoid the same ambiguity.

class C : I1, I2
{
   ...
   void X() 
   {  
     M();             // error, which one? 
     ((I1)this).M();  // OK 
   }
}

As said in comments, you have placed public constructor to the OpenExe method. That is just wrong construct and that is the reason why your code does not compile.
Move that constructor out of the OpenExe, as well as ‘SplashStart’.
Note that you already have another public default constructor, delete that one first.
You might also want to call that constructor from the other one, as this one is showing your splashscreen and doing some initalization.

public Form1(ApplicationControl appControl, MenuStrip, ...) :this()
{
    // your other initialization code
}

And another piece of advice, if you allow, do some serious refactoring here.
A method (or constructor) should not have more then seven parameters.
Consider to extract some objects, grouping similar elements, like ToolStripMenuItem, for example.

Also, creating a separate thread in the form constructor and putting constructor body to sleep might not be the best practice as well.
Create your form, just don’t show it. You can, for example, show splashscreen, initialize a timer. Once 5 seconds is over, hide splashscreen, show your main form.
Also, t.Abort(); is not recommended as well.
Good luck. Hope this helps you a bit.

As said in comments, you have placed public constructor to the OpenExe method. That is just wrong construct and that is the reason why your code does not compile.
Move that constructor out of the OpenExe, as well as ‘SplashStart’.
Note that you already have another public default constructor, delete that one first.
You might also want to call that constructor from the other one, as this one is showing your splashscreen and doing some initalization.

public Form1(ApplicationControl appControl, MenuStrip, ...) :this()
{
    // your other initialization code
}

And another piece of advice, if you allow, do some serious refactoring here.
A method (or constructor) should not have more then seven parameters.
Consider to extract some objects, grouping similar elements, like ToolStripMenuItem, for example.

Also, creating a separate thread in the form constructor and putting constructor body to sleep might not be the best practice as well.
Create your form, just don’t show it. You can, for example, show splashscreen, initialize a timer. Once 5 seconds is over, hide splashscreen, show your main form.
Also, t.Abort(); is not recommended as well.
Good luck. Hope this helps you a bit.

При использовании явной реализации интерфейса члены в принудительном порядке вынуждены использовать что-то более ограниченное, чем private. И когда модификатор доступа принудительно, вы можете его не добавлять.

Аналогично, в самом интерфейсе все члены public. Если вы попытаетесь добавить модификатор внутри интерфейса, вы получите аналогичную ошибку.

Почему явные члены (очень) частные? Рассмотрим:

interface I1 { void M(); }
interface I2 { void M(); }

class C : I1, I2
{
    void I1.M() { ... }
    void I2.M() { ... }
}

C c = new C();
c.M();         // Error, otherwise: which one?
(c as I1).M(); // Ok, no ambiguity. 

Если эти методы были общедоступными, у вас будет столкновение имен, которое не может быть разрешено нормальными правилами перегрузки.

По той же причине вы даже не можете вызвать M() изнутри члена class C. Перед тем, как избежать одной и той же двусмысленности, вам придется отбрасывать this на определенный интерфейс.

class C : I1, I2
{
   ...
   void X() 
   {  
     M();             // error, which one? 
     ((I1)this).M();  // OK 
   }
}