setState doesn't update the state immediately

Your state needs some time to mutate, and since console.log(this.state.boardAddModalShow) executes before the state mutates, you get the previous value as output. So you need to write the console in the callback to the setState function

openAddBoardModal() {
  this.setState({ boardAddModalShow: true }, function () {
    console.log(this.state.boardAddModalShow);
  });
}

setState is asynchronous. It means you can’t call it on one line and assume the state has changed on the next.

According to React docs

setState() does not immediately mutate this.state but creates a pending state transition. Accessing this.state after calling this method can potentially return the existing value. There is no guarantee of synchronous operation of calls to setState and calls may be batched for performance gains.

Why would they make setState async

This is because setState alters the state and causes rerendering. This can be an expensive operation and making it synchronous might leave the browser unresponsive.

Thus the setState calls are asynchronous as well as batched for better UI experience and performance.

Fortunately setState() takes a callback. And this is where we get updated state.

Consider this example.

this.setState({ name: "myname" }, () => {                              
        //callback
        console.log(this.state.name) // myname
      });

So When callback fires, this.state is the updated state.
You can get mutated/updated data in callback.

Since setSatate is a asynchronous function so you need to console the state as a callback like this.

openAddBoardModal(){
    this.setState({ boardAddModalShow: true }, () => {
        console.log(this.state.boardAddModalShow)
    });
}

For anyone trying to do this with hooks, you need useEffect.

function App() {
  const [x, setX] = useState(5)
  const [y, setY] = useState(15) 

  console.log("Element is rendered:", x, y)

  // setting y does not trigger the effect
  // the second argument is an array of dependencies
  useEffect(() => console.log("re-render because x changed:", x), [x])

  function handleXClick() {
    console.log("x before setting:", x)
    setX(10)
    console.log("x in *line* after setting:", x)
  }

  return <>
    <div> x is {x}. </div>
    <button onClick={handleXClick}> set x to 10</button>
    <div> y is {y}. </div>
    <button onClick={() => setY(20)}> set y to 20</button>
  </>
}

Output:

Element is rendered: 5 15
re-render because x changed: 5
(press x button)
x before setting: 5
x in *line* after setting: 5
Element is rendered: 10 15
re-render because x changed: 10
(press y button)
Element is rendered: 10 20