You may need a number line to help you, or if you understand how to use a thermometer, that will help.
Your number line is like a thermometer. Somewhere round the middle of the line mark 0. To the left are all the negative numbers, to the right all the positives. On a thermometer (Celsius) 0 is the freezing point. All to the left are below freezing; all to the right are above freezing.
-4 and -9 are on the left of zero, and -9 is further to the left of zero than -4, which means -9 is to the left of -4. Now take a positive number, say, 5 to the right of zero. What is 5-(-4)? To find out you want to know the distance between them on the number line. The answer is 9, because the distance between zero and 5 is 5 and the distance between -4 and zero is 4; the distance between -4 and 5 is therefore 9. So 5-(-4)=5+4=9.
Remember, negative is left and positive is right. What is -1-(-4)? What does it mean? We start at -1, which is 1 to the left of zero. How far away is -4? It's 3 further to the left. But we know from above that 5-(-4) is the same as 5+4, so -1-(-4) is -1+4=4-1=3. So when we write -1-(-4), we're really saying how far is -1 away from -4. First thing to note is that it's to the right of 4, so right means positive so we know the answer is that -1 and -4 are separated by 3, and -1 is bigger than -4 by (positive) 3. What about -4-(-1)? To get from -1 to -4 we have to move 3 to the left so the answer is -3, because left is negative.
-4+(-9) means start at -4 and go another 9 places to the left, ending up at -13. -4+(-9) is the same as -4-9.
Play around with these ideas, and think also about thermometers and temperatures. -4 is a lower temperature than -1, and they're both below freezing. The better you can visualise numbers and abstract concepts the easier you will find the math.