How do I correctly use enum within a class?
I see two separate issues here. 

First of all your enum isn't available as a variable. 
By declaring it as a private type within your class the enum can't be used outside of that class.

This forces a user of that class to handle unintended calls like (attention, bug):

CandleRack(255);


which can make code harder to read (and debugging more difficult, since it wouldn't be detected by the compiler).

Second of all you try to directly assign a byte value to an enum type. This will be rejected by the compiler and/or cause runtime errors later (depending on compiler settings).

I recommend that you declare the enum type so that it is accessible directly via the header file:

.h

enum CandleRackStatus {
    inactive = 0,
    candlesOn = 1,
    rewardAnimation = 2,
    tweetArrived = 3,
    partyMode = 4,      
};

class CandleRack
{
  public:

    CandleRack(CandleRackStatus candleRackStatus);
    ~CandleRack();
    void begin();
    void pushButton();
    void selectRandomCandle();
    void turnOnCandle(byte candlePosition);
    void burnCandlesForMs();
    void fadeOutCandle(byte candlePosition);

  private:
    Bounce *_debouncePushButton;
    CandleRackStatus _candleRackStatus;
    byte _pushButtonPin; 
    byte _candlePosition;
    byte _candleStatus[NUM_LEDS];
    elapsedMillis _candleTimeElapsed[NUM_LEDS];
    unsigned int _candleOnForMs;
};


Now you can instantiate your class while using the correct enum type as the parameter.

.ino

CandleRack MyCandleRack(CandleRackStatus::inactive);


Within your implementation you can now directly store the constructor parameter in the private variable, the compiler will make sure it's always of the correct enum type:

.cpp

CandleRack::CandleRack(CandleRackStatus candleRackStatus) {
    _candleRackStatus = candleRackStatus;
    // rest of your initialisiation goes here
}


As an added bonus, there's the possibility to use a couple of C++ language features that will make your implementation easier:


a default parameter value
an initializer (list)


.h

class 
{
  public:
    CandleRack(CandleRackStatus candleRackStatus = CandleRackStatus::inactive);
    // [...]
}


.cpp

CandleRack::CandleRack(CandleRackStatus candleRackStatus) : _candleRackStatus(candleRackStatus) {
    // rest of your initializiation goes here
}