How do I print out the value of this boolean? (Java)

you should just remove the 'boolean' in front of your boolean variable.

Do it like this:

boolean isLeapYear = true;
System.out.println(isLeapYear);

or

boolean isLeapYear = true;
System.out.println(isLeapYear?"yes":"no");

The other thing ist hat you seems not to call the method at all! The method and the variable are both not static, thus, you have to create an instance of your class first. Or you just make both static and than simply call your method directly from your maim method.

Thus there are a couple of mistakes in the code. May be you shoud start with a more simple example and than rework it until it does what you want.

Example:

import java.util.Scanner;

public class booleanfun    {
    static boolean isLeapYear;

    public static void main(String[] args)
    {
        System.out.println("Enter a year to determine if it is a leap year or not: ");
        Scanner kboard = new Scanner(System.in);
        int year = kboard.nextInt();
        isLeapYear(year);
    }
    public static boolean isLeapYear(int year) {
        if (year % 4 != 0)
        isLeapYear = false;

        else if ((year % 4 == 0) && (year % 100 == 0))

        isLeapYear = false;

        else if ((year % 4 == 0) && (year % 100 == 0) && (year % 400 == 0))
            isLeapYear = true;

        else
            isLeapYear = false;

        System.out.println(isLeapYear);

        return isLeapYear;
    }
}

There are several problems.

One is of style; always capitalize class names. This is a universally observed Java convention. Failing to do so confuses other programmers.

Secondly, the line

System.out.println(boolean isLeapYear);

is a syntax error. Delete it.

Thirdly.

You never call the function from your main routine. That is why you never see any reply to the input.

There are a couple of ways to address your problem, however this is probably the most straightforward:

Your main method is static, so it does not have access to instance members (isLeapYear field and isLeapYear method. One approach to rectify this is to make both the field and the method static as well:

static boolean isLeapYear;
/* (snip) */
public static boolean isLeapYear(int year)
{
  /* (snip) */
}

Lastly, you're not actually calling your isLeapYear method (which is why you're not seeing any results). Add this line after int year = kboard.nextInt();:

isLeapYear(year);

That should be a start. There are some other best practices you could follow but for now just focus on getting your code to work; you can refactor later.

System.out.println(isLeapYear);

should work just fine.

Incidentally, in

else if ((year % 4 == 0) && (year % 100 == 0))
    isLeapYear = false;

else if ((year % 4 == 0) && (year % 100 == 0) && (year % 400 == 0))
    isLeapYear = true;

the year % 400 part will never be reached because if (year % 4 == 0) && (year % 100 == 0) && (year % 400 == 0) is true, then (year % 4 == 0) && (year % 100 == 0) must have succeeded.

Maybe swap those two conditions or refactor them:

else if ((year % 4 == 0) && (year % 100 == 0))
    isLeapYear = (year % 400 == 0);